Formula for the roots of a quadratic equation through the discriminant. Calculation options for securing material
Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factoring a quadratic trinomial. Geometric interpretation. Examples of determining roots and factoring.
Basic formulas
Consider the quadratic equation:
(1)
.
Roots of a quadratic equation(1) are determined by the formulas:
;
.
These formulas can be combined like this:
.
When the roots of a quadratic equation are known, then a polynomial of the second degree can be represented as a product of factors (factored):
.
Next we assume that are real numbers.
Let's consider discriminant of a quadratic equation:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
;
.
Then the factorization of the quadratic trinomial has the form:
.
If the discriminant is equal to zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is the imaginary unit, ;
and are the real and imaginary parts of the roots:
;
.
Then
.
Graphic interpretation
If you build graph of a function
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
At , the graph intersects the x-axis (axis) at two points.
When , the graph touches the x-axis at one point.
When , the graph does not cross the x-axis.
Below are examples of such graphs.
Useful formulas related to quadratic equation
(f.1) ;
(f.2) ;
(f.3) .
Derivation of the formula for the roots of a quadratic equation
We carry out transformations and apply formulas (f.1) and (f.3):
,
Where
;
.
So, we got the formula for a polynomial of the second degree in the form:
.
This shows that the equation
performed at
And .
That is, and are the roots of the quadratic equation
.
Examples of determining the roots of a quadratic equation
Example 1
(1.1)
.
Solution
.
Comparing with our equation (1.1), we find the values of the coefficients:
.
We find the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.
From this we obtain the factorization of the quadratic trinomial:
.
Graph of the function y = 2 x 2 + 7 x + 3 intersects the x-axis at two points.
Let's plot the function
.
The graph of this function is a parabola. It crosses the abscissa axis (axis) at two points:
And .
These points are the roots of the original equation (1.1).
Answer
;
;
.
Example 2
Find the roots of a quadratic equation:
(2.1)
.
Solution
Let's write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values of the coefficients:
.
We find the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.
Then the factorization of the trinomial has the form:
.
Graph of the function y = x 2 - 4 x + 4 touches the x-axis at one point.
Let's plot the function
.
The graph of this function is a parabola. It touches the x-axis (axis) at one point:
.
This point is the root of the original equation (2.1). Because this root is factored twice:
,
then such a root is usually called a multiple. That is, they believe that there are two equal roots:
.
Answer
;
.
Example 3
Find the roots of a quadratic equation:
(3.1)
.
Solution
Let's write the quadratic equation in general form:
(1)
.
Let's rewrite the original equation (3.1):
.
Comparing with (1), we find the values of the coefficients:
.
We find the discriminant:
.
The discriminant is negative, .
Therefore there are no real roots.
;
;
.
You can find complex roots:
.
Then
Let's plot the function
.
The graph of the function does not cross the x-axis. There are no real roots.
Answer
The graph of this function is a parabola. It does not intersect the x-axis (axis). Therefore there are no real roots.
;
;
.
There are no real roots. Complex roots:
I hope that after studying this article you will learn how to find the roots of a complete quadratic equation. Using the discriminant, only complete quadratic equations are solved; for solving incomplete ones quadratic equations
use other methods that you will find in the article "Solving incomplete quadratic equations". What quadratic equations are called complete? This equations of the form ax 2 + b x + c = 0
, where coefficients a, b and c are not equal to zero. So, to solve a complete quadratic equation, we need to calculate the discriminant D.
D = b 2 – 4ac.
Depending on the value of the discriminant, we will write down the answer.< 0),то корней нет.
If the discriminant is a negative number (D
If the discriminant is zero, then x = (-b)/2a. When the discriminant is a positive number (D > 0),
then x 1 = (-b - √D)/2a, and x 2 = (-b + √D)/2a. For example. Solve the equation x 2
– 4x + 4= 0.
D = 4 2 – 4 4 = 0
x = (- (-4))/2 = 2
Answer: 2. Solve the equation Solve Equation 2
+ x + 3 = 0.
D = 1 2 – 4 2 3 = – 23.
Answer: 2. Solve the equation Answer: no roots.
+ 5x – 7 = 0
D = 5 2 – 4 2 (–7) = 81
x 1 = (-5 - √81)/(2 2)= (-5 - 9)/4= – 3.5
x 2 = (-5 + √81)/(2 2) = (-5 + 9)/4=1.
Answer: – 3.5; 1
Using these formulas you can solve any complete quadratic equation. You just need to be careful to the equation was written as a polynomial of the standard form
A Solve the equation + bx + c, otherwise you may make a mistake. For example, in writing the equation x + 3 + 2x 2 = 0, you can mistakenly decide that
a = 1, b = 3 and c = 2. Then
D = 3 2 – 4 1 2 = 1 and then the equation has two roots. And this is not true. (See solution to example 2 above).
Therefore, if the equation is not written as a polynomial of the standard form, first the complete quadratic equation must be written as a polynomial of the standard form (the monomial with the largest exponent should come first, that is A Solve the equation , then with less – bx and then a free member With.
When solving the reduced quadratic equation and a quadratic equation with an even coefficient in the second term, you can use other formulas. Let's get acquainted with these formulas. If in a complete quadratic equation the second term has an even coefficient (b = 2k), then you can solve the equation using the formulas shown in the diagram in Figure 2.
A complete quadratic equation is called reduced if the coefficient at Solve the equation is equal to one and the equation takes the form x 2 + px + q = 0. Such an equation can be given for solution, or it can be obtained by dividing all coefficients of the equation by the coefficient A, standing at Solve the equation .
Figure 3 shows a diagram for solving the reduced square 
equations. Let's look at an example of the application of the formulas discussed in this article.
Example. Solve the equation
3Solve the equation + 6x – 6 = 0.
Let's solve this equation using the formulas shown in the diagram in Figure 1.
D = 6 2 – 4 3 (– 6) = 36 + 72 = 108
√D = √108 = √(36 3) = 6√3
x 1 = (-6 - 6√3)/(2 3) = (6 (-1- √(3)))/6 = –1 – √3
x 2 = (-6 + 6√3)/(2 3) = (6 (-1+ √(3)))/6 = –1 + √3
Answer: –1 – √3; –1 + √3
You can notice that the coefficient of x in this equation is an even number, that is, b = 6 or b = 2k, whence k = 3. Then let’s try to solve the equation using the formulas shown in the diagram of the figure D 1 = 3 2 – 3 · (– 6 ) = 9 + 18 = 27
√(D 1) = √27 = √(9 3) = 3√3
x 1 = (-3 - 3√3)/3 = (3 (-1 - √(3)))/3 = – 1 – √3
x 2 = (-3 + 3√3)/3 = (3 (-1 + √(3)))/3 = – 1 + √3
Answer: –1 – √3; –1 + √3. Noticing that all the coefficients in this quadratic equation are divisible by 3 and performing the division, we get the reduced quadratic equation x 2 + 2x – 2 = 0 Solve this equation using the formulas for the reduced quadratic 
equations figure 3.
D 2 = 2 2 – 4 (– 2) = 4 + 8 = 12
√(D 2) = √12 = √(4 3) = 2√3
x 1 = (-2 - 2√3)/2 = (2 (-1 - √(3)))/2 = – 1 – √3
x 2 = (-2 + 2√3)/2 = (2 (-1+ √(3)))/2 = – 1 + √3
Answer: –1 – √3; –1 + √3.
As we see, when solving this equation by various formulas we received the same answer. Therefore, having thoroughly mastered the formulas shown in the diagram in Figure 1, you will always be able to solve any complete quadratic equation.
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IN modern society the ability to perform operations with equations containing a variable squared can be useful in many areas of activity and is widely used in practice in scientific and technical developments. Evidence of this can be found in the design of sea and river vessels, aircraft and missiles. Using such calculations, the trajectories of movement of a wide variety of bodies, including space objects, are determined. Examples with the solution of quadratic equations are used not only in economic forecasting, in the design and construction of buildings, but also in the most ordinary everyday circumstances. They may be needed in hiking trips, at sporting events, in stores while shopping, and in other very common situations.
Let's break the expression into its component factors
The degree of an equation is determined by the maximum value of the degree of the variable that the expression contains. If it is equal to 2, then such an equation is called quadratic.
If we speak in the language of formulas, then the indicated expressions, no matter how they look, can always be brought to the form when the left side of the expression consists of three terms. Among them: ax 2 (that is, a variable squared with its coefficient), bx (an unknown without a square with its coefficient) and c (a free component, that is, an ordinary number). All this on the right side is equal to 0. In the case when such a polynomial lacks one of its constituent terms, with the exception of ax 2, it is called an incomplete quadratic equation. Examples with the solution of such problems, the values of the variables in which are easy to find, should be considered first.
If the expression looks like it has two terms on the right side, more precisely ax 2 and bx, the easiest way to find x is by putting the variable out of brackets. Now our equation will look like this: x(ax+b). Next, it becomes obvious that either x=0, or the problem comes down to finding a variable from the following expression: ax+b=0. This is dictated by one of the properties of multiplication. The rule states that the product of two factors results in 0 only if one of them is zero.
Example
x=0 or 8x - 3 = 0
As a result, we get two roots of the equation: 0 and 0.375.
Equations of this kind can describe the movement of bodies under the influence of gravity, which began to move from a certain point taken as the origin of coordinates. Here mathematical notation takes the following form: y = v 0 t + gt 2 /2. By substituting the necessary values, equating the right side to 0 and finding possible unknowns, you can find out the time that passes from the moment the body rises to the moment it falls, as well as many other quantities. But we'll talk about this later.
Factoring an Expression
The rule described above makes it possible to solve these problems in more complex cases. Let's look at examples of solving quadratic equations of this type.
X 2 - 33x + 200 = 0
This quadratic trinomial is complete. First, let's transform the expression and factor it. There are two of them: (x-8) and (x-25) = 0. As a result, we have two roots 8 and 25.
Examples with solving quadratic equations in grade 9 allow this method to find a variable in expressions not only of the second, but even of the third and fourth orders.
For example: 2x 3 + 2x 2 - 18x - 18 = 0. When factoring the right side into factors with a variable, there are three of them, that is, (x+1), (x-3) and (x+3).
As a result, it becomes obvious that given equation has three roots: -3; -1; 3.
Square Root
Another case of an incomplete second-order equation is an expression represented in the language of letters in such a way that right part is constructed from the components ax 2 and c. Here, to obtain the value of the variable, the free term is transferred to the right side, and after that, it is extracted from both sides of the equality Square root. It should be noted that in this case there are usually two roots of the equation. The only exceptions can be equalities that do not contain a term with at all, where the variable is equal to zero, as well as variants of expressions when the right side is negative. In the latter case, there are no solutions at all, since the above actions cannot be performed with roots. Examples of solutions to quadratic equations of this type should be considered.
In this case, the roots of the equation will be the numbers -4 and 4.
Calculation of land area
The need for this kind of calculations appeared in ancient times, because the development of mathematics in those distant times was largely determined by the need to determine with the greatest accuracy the areas and perimeters of land plots.
We should also consider examples of solving quadratic equations based on problems of this kind.
So, let's say there is a rectangular plot of land, the length of which is 16 meters greater than the width. You should find the length, width and perimeter of the site if you know that its area is 612 m2.
To get started, let's first create the necessary equation. Let us denote by x the width of the area, then its length will be (x+16). From what has been written it follows that the area is determined by the expression x(x+16), which, according to the conditions of our problem, is 612. This means that x(x+16) = 612.
Solving complete quadratic equations, and this expression is exactly that, cannot be done in the same way. Why? Although the left side still contains two factors, their product does not equal 0 at all, so different methods are used here.
Discriminant
First of all, let's make the necessary transformations, then appearance of this expression will look like this: x 2 + 16x - 612 = 0. This means that we have received an expression in a form corresponding to the previously specified standard, where a=1, b=16, c=-612.
This could be an example of solving quadratic equations using a discriminant. Here the necessary calculations are made according to the scheme: D = b 2 - 4ac. This auxiliary quantity not only makes it possible to find the required quantities in a second-order equation, it determines the number of possible options. If D>0, there are two of them; for D=0 there is one root. In case D<0, никаких шансов для решения у уравнения вообще не имеется.
About roots and their formula
In our case, the discriminant is equal to: 256 - 4(-612) = 2704. This suggests that our problem has an answer. If you know k, the solution of quadratic equations must be continued using the formula below. It allows you to calculate the roots.
This means that in the presented case: x 1 =18, x 2 =-34. The second option in this dilemma cannot be a solution, because the dimensions of the land plot cannot be measured in negative quantities, which means x (that is, the width of the plot) is 18 m. From here we calculate the length: 18+16=34, and the perimeter 2(34+ 18)=104(m2).
Examples and tasks
We continue our study of quadratic equations. Examples and detailed solutions of several of them will be given below.
1) 15x 2 + 20x + 5 = 12x 2 + 27x + 1
Let’s move everything to the left side of the equality, make a transformation, that is, we’ll get the type of equation that is usually called standard, and equate it to zero.
15x 2 + 20x + 5 - 12x 2 - 27x - 1 = 0
Adding similar ones, we determine the discriminant: D = 49 - 48 = 1. This means our equation will have two roots. Let's calculate them according to the above formula, which means that the first of them will be equal to 4/3, and the second to 1.
2) Now let's solve mysteries of a different kind.
Let's find out if there are any roots here x 2 - 4x + 5 = 1? To obtain a comprehensive answer, let’s reduce the polynomial to the corresponding usual form and calculate the discriminant. In the above example, it is not necessary to solve the quadratic equation, because this is not the essence of the problem at all. In this case, D = 16 - 20 = -4, which means there really are no roots.
Vieta's theorem
It is convenient to solve quadratic equations using the above formulas and the discriminant, when the square root is taken from the value of the latter. But this does not always happen. However, there are many ways to obtain the values of variables in this case. Example: solving quadratic equations using Vieta's theorem. She is named after who lived in the 16th century in France and made a brilliant career thanks to his mathematical talent and connections at court. His portrait can be seen in the article.
The pattern that the famous Frenchman noticed was as follows. He proved that the roots of the equation add up numerically to -p=b/a, and their product corresponds to q=c/a.
Now let's look at specific tasks.
3x 2 + 21x - 54 = 0
For simplicity, let's transform the expression:
x 2 + 7x - 18 = 0
Let's use Vieta's theorem, this will give us the following: the sum of the roots is -7, and their product is -18. From here we get that the roots of the equation are the numbers -9 and 2. After checking, we will make sure that these variable values really fit into the expression.
Parabola graph and equation
The concepts of quadratic function and quadratic equations are closely related. Examples of this have already been given earlier. Now let's look at some mathematical riddles in a little more detail. Any equation of the described type can be represented visually. Such a relationship, drawn as a graph, is called a parabola. Its various types are presented in the figure below.
Any parabola has a vertex, that is, a point from which its branches emerge. If a>0, they go high to infinity, and when a<0, они рисуются вниз. Простейшим примером подобной зависимости является функция y = x 2 . В данном случае в уравнении x 2 =0 неизвестное может принимать только одно значение, то есть х=0, а значит существует только один корень. Это неудивительно, ведь здесь D=0, потому что a=1, b=0, c=0. Выходит формула корней (точнее одного корня) квадратного уравнения запишется так: x = -b/2a.
Visual representations of functions help solve any equations, including quadratic ones. This method is called graphical. And the value of the x variable is the abscissa coordinate at the points where the graph line intersects with 0x. The coordinates of the vertex can be found using the formula just given x 0 = -b/2a. And by substituting the resulting value into the original equation of the function, you can find out y 0, that is, the second coordinate of the vertex of the parabola, which belongs to the ordinate axis.
The intersection of the branches of a parabola with the abscissa axis
There are a lot of examples of solving quadratic equations, but there are also general patterns. Let's look at them. It is clear that the intersection of the graph with the 0x axis for a>0 is possible only if 0 takes negative values. And for a<0 координата у 0 должна быть положительна. Для указанных вариантов D>0. Otherwise D<0. А когда D=0, вершина параболы расположена непосредственно на оси 0х.
From the graph of the parabola you can also determine the roots. The opposite is also true. That is, if it is not easy to obtain a visual representation of a quadratic function, you can equate the right side of the expression to 0 and solve the resulting equation. And knowing the points of intersection with the 0x axis, it is easier to construct a graph.
From the history
Using equations containing a squared variable, in the old days they not only made mathematical calculations and determined the areas of geometric figures. The ancients needed such calculations for grand discoveries in the fields of physics and astronomy, as well as for making astrological forecasts.
As modern scientists suggest, the inhabitants of Babylon were among the first to solve quadratic equations. This happened four centuries before our era. Of course, their calculations were radically different from those currently accepted and turned out to be much more primitive. For example, Mesopotamian mathematicians had no idea about the existence of negative numbers. They were also unfamiliar with other subtleties that any modern schoolchild knows.
Perhaps even earlier than the scientists of Babylon, the sage from India Baudhayama began solving quadratic equations. This happened about eight centuries before the era of Christ. True, the second-order equations, the methods for solving which he gave, were the simplest. Besides him, Chinese mathematicians were also interested in similar questions in the old days. In Europe, quadratic equations began to be solved only at the beginning of the 13th century, but later they were used in their works by such great scientists as Newton, Descartes and many others.
Quadratic equations. Discriminant. Solution, examples.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Types of quadratic equations
What is a quadratic equation? What does it look like? In term quadratic equation the keyword is "square". This means that in the equation Necessarily there must be an x squared. In addition to it, the equation may (or may not!) contain just X (to the first power) and just a number (free member). And there should be no X's to a power greater than two.
In mathematical terms, a quadratic equation is an equation of the form:
Here a, b and c- some numbers. b and c- absolutely any, but A– anything other than zero. For example:
Here A =1; b = 3; c = -4
Here A =2; b = -0,5; c = 2,2
Here A =-3; b = 6; c = -18
Well, you understand...
In these quadratic equations on the left there is full set members. X squared with a coefficient A, x to the first power with coefficient b And free member s.
Such quadratic equations are called full.
And if b= 0, what do we get? We have X will be lost to the first power. This happens when multiplied by zero.) It turns out, for example:
5x 2 -25 = 0,
2x 2 -6x=0,
-x 2 +4x=0
And so on. And if both coefficients b And c are equal to zero, then it’s even simpler:
2x 2 =0,
-0.3x 2 =0
Such equations where something is missing are called incomplete quadratic equations. Which is quite logical.) Please note that x squared is present in all equations.
By the way, why A can't be equal to zero? And you substitute instead A zero.) Our X squared will disappear! The equation will become linear. And the solution is completely different...
That's all the main types of quadratic equations. Complete and incomplete.
Solving quadratic equations.
Solving complete quadratic equations.
Quadratic equations are easy to solve. According to formulas and clear, simple rules. At the first stage, it is necessary to bring the given equation to a standard form, i.e. to the form:
If the equation is already given to you in this form, you do not need to do the first stage.) The main thing is to correctly determine all the coefficients, A, b And c.
The formula for finding the roots of a quadratic equation looks like this:
The expression under the root sign is called discriminant. But more about him below. As you can see, to find X, we use only a, b and c. Those. coefficients from a quadratic equation. Just carefully substitute the values a, b and c We calculate into this formula. Let's substitute with your own signs! For example, in the equation:
A =1; b = 3; c= -4. Here we write it down:
The example is almost solved:
This is the answer.
Everything is very simple. And what, you think it’s impossible to make a mistake? Well, yes, how...
The most common mistakes are confusion with sign values a, b and c. Or rather, not with their signs (where to get confused?), but with the substitution of negative values into the formula for calculating the roots. What helps here is a detailed recording of the formula with specific numbers. If there are problems with calculations, do that!
Suppose we need to solve the following example:
Here a = -6; b = -5; c = -1
Let's say you know that you rarely get answers the first time.
Well, don't be lazy. It will take about 30 seconds to write an extra line. And the number of errors will decrease sharply. So we write in detail, with all the brackets and signs:
It seems incredibly difficult to write out so carefully. But it only seems so. Give it a try. Well, or choose. What's better, fast or right?
Besides, I will make you happy. After a while, there will be no need to write everything down so carefully. It will turn out right on its own. Especially if you use practical techniques that are described below. This evil example with a bunch of minuses can be solved easily and without errors!
But, often, quadratic equations look slightly different. For example, like this: Did you recognize it?) Yes! This.
incomplete quadratic equations
Solving incomplete quadratic equations. a, b and c.
They can also be solved using a general formula. You just need to understand correctly what they are equal to here. Have you figured it out? In the first example a = 1; b = -4; c A ? It's not there at all! Well yes, that's right. In mathematics this means that c = 0 ! That's all. Substitute zero into the formula instead c, and we will succeed. Same with the second example. Only we don’t have zero here With b !
But incomplete quadratic equations can be solved much more simply. Without any formulas. Let's consider the first incomplete equation. What can you do on the left side? You can take X out of brackets! Let's take it out.
And what from this? And the fact that the product equals zero if and only if any of the factors equals zero! Don't believe me? Okay, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? That's it...
Therefore, we can confidently write: x 1 = 0, x 2 = 4.
All. These will be the roots of our equation. Both are suitable. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much simpler than using the general formula. Let me note, by the way, which X will be the first and which will be the second - absolutely indifferent. It is convenient to write in order, x 1- what is smaller and Solve the equation- that which is greater.
The second equation can also be solved simply. Move 9 to the right side. We get:
All that remains is to extract the root from 9, and that’s it. It will turn out:
Also two roots . x 1 = -3, x 2 = 3.
This is how all incomplete quadratic equations are solved. Either by placing X out of brackets, or by simply moving the number to the right and then extracting the root.
It is extremely difficult to confuse these techniques. Simply because in the first case you will have to extract the root of X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets...
Discriminant. Discriminant formula.
Magic word discriminant ! Rarely a high school student has not heard this word! The phrase “we solve through a discriminant” inspires confidence and reassurance. Because there is no need to expect tricks from the discriminant! It is simple and trouble-free to use.) I remind you of the most general formula for solving any quadratic equations:
The expression under the root sign is called a discriminant. Typically the discriminant is denoted by the letter D. Discriminant formula:
D = b 2 - 4ac
And what is so remarkable about this expression? Why did it deserve a special name? What the meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically call it anything... Letters and letters.
Here's the thing. When solving a quadratic equation using this formula, it is possible only three cases.
1. The discriminant is positive. This means the root can be extracted from it. Whether the root is extracted well or poorly is another question. What is important is what is extracted in principle. Then your quadratic equation has two roots. Two different solutions.
2. The discriminant is zero. Then you will have one solution. Since adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not one root, but two identical. But, in a simplified version, it is customary to talk about one solution.
3. The discriminant is negative. The square root of a negative number cannot be taken. Well, okay. This means there are no solutions.
To be honest, when simply solving quadratic equations, the concept of a discriminant is not really needed. We substitute the values of the coefficients into the formula and count. Everything happens there by itself, two roots, one, and none. However, when solving more complex tasks, without knowledge meaning and formula of the discriminant not enough. Especially in equations with parameters. Such equations are aerobatics for the State Examination and the Unified State Examination!)
So, how to solve quadratic equations through the discriminant you remembered. Or you learned, which is also not bad.) You know how to correctly determine a, b and c. Do you know how? attentively substitute them into the root formula and attentively count the result. You understand that the key word here is attentively?
Now take note of practical techniques that dramatically reduce the number of errors. The same ones that are due to inattention... For which it later becomes painful and offensive...
First appointment
. Don’t be lazy before solving a quadratic equation and bring it to standard form. What does this mean?
Let's say that after all the transformations you get the following equation:
Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c. Construct the example correctly. First, X squared, then without square, then the free term. Like this:
And again, don’t rush! A minus in front of an X squared can really upset you. It's easy to forget... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the entire equation by -1. We get:
But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example. Decide for yourself.
You should now have roots 2 and -1. Reception second. Check the roots! According to Vieta's theorem. Don't be afraid, I'll explain everything! Checking last thing the equation. Those. the one we used to write down the root formula. If (as in this example) the coefficient a = 1 , checking the roots is easy. It is enough to multiply them. The result should be a free member, i.e. in our case -2. Please note, not 2, but -2! Free member with your sign
. If it doesn’t work out, it means they’ve already screwed up somewhere. Look for the error. b If it works, you need to add the roots. Last and final check. The coefficient should be With
opposite b familiar. In our case -1+2 = +1. A coefficient
It’s a pity that this is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! There will be fewer and fewer errors.
Reception third . If your equation has fractional coefficients, get rid of the fractions! Multiply the equation by a common denominator as described in the lesson "How to solve equations? Identity transformations." When working with fractions, errors keep creeping in for some reason...
By the way, I promised to simplify the evil example with a bunch of minuses. Please! Here he is.
In order not to get confused by the minuses, we multiply the equation by -1. We get:
That's all! Solving is a pleasure!
So, let's summarize the topic.
Practical tips:
1. Before solving, we bring the quadratic equation to standard form and build it Right.
2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying the entire equation by -1.
3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding factor.
4. If x squared is pure, its coefficient is equal to one, the solution can be easily verified using Vieta’s theorem. Do it!
Now we can decide.)
Solve equations:
8x 2 - 6x + 1 = 0
x 2 + 3x + 8 = 0
x 2 - 4x + 4 = 0
(x+1) 2 + x + 1 = (x+1)(x+2)
Answers (in disarray):
x 1 = 0
x 2 = 5
x 1.2 =2
x 1 = 2
x 2 = -0.5
x - any number
x 1 = -3
x 2 = 3
no solutions
x 1 = 0.25
x 2 = 0.5
Does everything fit? Great! Quadratic equations are not your headache. The first three worked, but the rest didn’t? Then the problem is not with quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.
Doesn't quite work out? Or does it not work out at all? Then Section 555 will help you. All these examples are broken down there. Shown main errors in the solution. Of course, we also talk about the use of identical transformations in solving various equations. Helps a lot!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
First level
Quadratic equations. The Comprehensive Guide (2019)
In the term “quadratic equation,” the key word is “quadratic.” This means that the equation must necessarily contain a variable (that same x) squared, and there should not be xes to the third (or greater) power.
The solution of many equations comes down to solving quadratic equations.
Let's learn to determine that this is a quadratic equation and not some other equation.
Example 1.
Let's get rid of the denominator and multiply each term of the equation by
Let's move everything to the left side and arrange the terms in descending order of powers of X
Now we can say with confidence that this equation is quadratic!
Example 2.
Multiply the left and right sides by:
This equation, although it was originally in it, is not quadratic!
Example 3.
Let's multiply everything by:
Scary? The fourth and second degrees... However, if we make a replacement, we will see that we have a simple quadratic equation:
Example 4.
It seems to be there, but let's take a closer look. Let's move everything to the left side:
See, it's reduced - and now it's a simple linear equation!
Now try to determine for yourself which of the following equations are quadratic and which are not:
Examples:
Answers:
- square;
- square;
- not square;
- not square;
- not square;
- square;
- not square;
- square.
Mathematicians conventionally divide all quadratic equations into the following types:
- Complete quadratic equations- equations in which the coefficients and, as well as the free term c, are not equal to zero (as in the example). In addition, among complete quadratic equations there are given- these are equations in which the coefficient (the equation from example one is not only complete, but also reduced!)
- Incomplete quadratic equations- equations in which the coefficient and or the free term c are equal to zero:
They are incomplete because they are missing some element. But the equation must always contain X squared!!! Otherwise, it will no longer be a quadratic equation, but some other equation.
Why did they come up with such a division? It would seem that there is an X squared, and okay. This division is determined by the solution methods. Let's look at each of them in more detail.
Solving incomplete quadratic equations
First, let's focus on solving incomplete quadratic equations - they are much simpler!
There are types of incomplete quadratic equations:
- , in this equation the coefficient is equal.
- , in this equation the free term is equal to.
- , in this equation the coefficient and the free term are equal.
1. i. Since we know how to take the square root, let's express from this equation
The expression can be either negative or positive. A squared number cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number, so: if, then the equation has no solutions.
And if, then we get two roots. These formulas do not need to be memorized. The main thing is that you must know and always remember that it cannot be less.
Let's try to solve some examples.
Example 5:
Solve the equation
Now all that remains is to extract the root from the left and right sides. After all, you remember how to extract roots?
Answer:
Never forget about roots with a negative sign!!!
Example 6:
Solve the equation
Answer:
Example 7:
Solve the equation
Oh! The square of a number cannot be negative, which means that the equation
no roots!
For such equations that have no roots, mathematicians came up with a special icon - (empty set). And the answer can be written like this:
Answer:
Thus, this quadratic equation has two roots. There are no restrictions here, since we did not extract the root.
Example 8:
Solve the equation
Let's take the common factor out of brackets:
Thus,
This equation has two roots.
Answer:
The simplest type of incomplete quadratic equations (although they are all simple, right?). Obviously, this equation always has only one root:
We will dispense with examples here.
Solving complete quadratic equations
We remind you that a complete quadratic equation is an equation of the form equation where
Solving complete quadratic equations is a little more difficult (just a little) than these.
Remember, Any quadratic equation can be solved using a discriminant! Even incomplete.
The other methods will help you do it faster, but if you have problems with quadratic equations, first master the solution using the discriminant.
1. Solving quadratic equations using a discriminant.
Solving quadratic equations using this method is very simple; the main thing is to remember the sequence of actions and a couple of formulas.
If, then the equation has a root. You need to pay special attention to the step. Discriminant () tells us the number of roots of the equation.
- If, then the formula in the step will be reduced to. Thus, the equation will only have a root.
- If, then we will not be able to extract the root of the discriminant at the step. This indicates that the equation has no roots.
Let's go back to our equations and look at some examples.
Example 9:
Solve the equation
Step 1 we skip.
Step 2.
We find the discriminant:
This means the equation has two roots.
Step 3.
Answer:
Example 10:
Solve the equation
The equation is presented in standard form, so Step 1 we skip.
Step 2.
We find the discriminant:
This means that the equation has one root.
Answer:
Example 11:
Solve the equation
The equation is presented in standard form, so Step 1 we skip.
Step 2.
We find the discriminant:
This means we will not be able to extract the root of the discriminant. There are no roots of the equation.
Now we know how to correctly write down such answers.
Answer: no roots
2. Solving quadratic equations using Vieta's theorem.
If you remember, there is a type of equation that is called reduced (when the coefficient a is equal to):
Such equations are very easy to solve using Vieta’s theorem:
Sum of roots given quadratic equation is equal, and the product of the roots is equal.
Example 12:
Solve the equation
This equation can be solved using Vieta's theorem because .
The sum of the roots of the equation is equal, i.e. we get the first equation:
And the product is equal to:
Let's compose and solve the system:
- And. The amount is equal to;
- And. The amount is equal to;
- And. The amount is equal.
and are the solution to the system:
Answer: ; .
Example 13:
Solve the equation
Answer:
Example 14:
Solve the equation
The equation is given, which means:
Answer:
QUADRATIC EQUATIONS. AVERAGE LEVEL
What is a quadratic equation?
In other words, a quadratic equation is an equation of the form, where - the unknown, - some numbers, and.
The number is called the highest or first coefficient quadratic equation, - second coefficient, A - free member.
Why? Because if the equation immediately becomes linear, because will disappear.
In this case, and can be equal to zero. In this chair equation is called incomplete. If all the terms are in place, that is, the equation is complete.
Solutions to various types of quadratic equations
Methods for solving incomplete quadratic equations:
First, let's look at methods for solving incomplete quadratic equations - they are simpler.
We can distinguish the following types of equations:
I., in this equation the coefficient and the free term are equal.
II. , in this equation the coefficient is equal.
III. , in this equation the free term is equal to.
Now let's look at the solution to each of these subtypes.
Obviously, this equation always has only one root:
A squared number cannot be negative, because when you multiply two negative or two positive numbers, the result will always be a positive number. That's why:
if, then the equation has no solutions;
if we have two roots
These formulas do not need to be memorized. The main thing to remember is that it cannot be less.
Examples:
Solutions:
Answer:
Never forget about roots with a negative sign!
The square of a number cannot be negative, which means that the equation
no roots.
To briefly write down that a problem has no solutions, we use the empty set icon.
Answer:
So, this equation has two roots: and.
Answer:
Let's take the common factor out of brackets:
The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:
So, this quadratic equation has two roots: and.
Example:
Solve the equation.
Solution:
Let's factor the left side of the equation and find the roots:
Answer:
Methods for solving complete quadratic equations:
1. Discriminant
Solving quadratic equations this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using a discriminant! Even incomplete.
Did you notice the root from the discriminant in the formula for roots? But the discriminant can be negative. What to do? We need to pay special attention to step 2. The discriminant tells us the number of roots of the equation.
- If, then the equation has roots:
- If, then the equation has the same roots, and in fact, one root:
Such roots are called double roots.
- If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.
Why are different numbers of roots possible? Let us turn to the geometric meaning of the quadratic equation. The graph of the function is a parabola:
In a special case, which is a quadratic equation, . This means that the roots of a quadratic equation are the points of intersection with the abscissa axis (axis). A parabola may not intersect the axis at all, or may intersect it at one (when the vertex of the parabola lies on the axis) or two points.
In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upward, and if, then downward.
Examples:
Solutions:
Answer:
Answer: .
Answer:
This means there are no solutions.
Answer: .
2. Vieta's theorem
It is very easy to use Vieta's theorem: you just need to choose a pair of numbers whose product is equal to the free term of the equation, and the sum is equal to the second coefficient taken with the opposite sign.
It is important to remember that Vieta's theorem can only be applied in reduced quadratic equations ().
Let's look at a few examples:
Example #1:
Solve the equation.
Solution:
This equation can be solved using Vieta's theorem because . Other coefficients: ; .
The sum of the roots of the equation is:
And the product is equal to:
Let's select pairs of numbers whose product is equal and check whether their sum is equal:
- And. The amount is equal to;
- And. The amount is equal to;
- And. The amount is equal.
and are the solution to the system:
Thus, and are the roots of our equation.
Answer: ; .
Example #2:
Solution:
Let's select pairs of numbers that give in the product, and then check whether their sum is equal:
and: they give in total.
and: they give in total. To obtain, it is enough to simply change the signs of the supposed roots: and, after all, the product.
Answer:
Example #3:
Solution:
The free term of the equation is negative, and therefore the product of the roots is a negative number. This is only possible if one of the roots is negative and the other is positive. Therefore the sum of the roots is equal to differences of their modules.
Let us select such pairs of numbers that give in the product, and the difference of which is equal to:
and: their difference is equal - does not fit;
and: - not suitable;
and: - not suitable;
and: - suitable. All that remains is to remember that one of the roots is negative. Since their sum must be equal, the root with a smaller modulus must be negative: . We check:
Answer:
Example #4:
Solve the equation.
Solution:
The equation is given, which means:
The free term is negative, and therefore the product of the roots is negative. And this is only possible when one root of the equation is negative and the other is positive.
Let's select pairs of numbers whose product is equal, and then determine which roots should have a negative sign:
Obviously, only the roots and are suitable for the first condition:
Answer:
Example #5:
Solve the equation.
Solution:
The equation is given, which means:
The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, it means both roots have a minus sign.
Let us select pairs of numbers whose product is equal to:
Obviously, the roots are the numbers and.
Answer:
Agree, it’s very convenient to come up with roots orally, instead of counting this nasty discriminant. Try to use Vieta's theorem as often as possible.
But Vieta’s theorem is needed in order to facilitate and speed up finding the roots. In order for you to benefit from using it, you must bring the actions to automaticity. And for this, solve five more examples. But don't cheat: you can't use a discriminant! Only Vieta's theorem:
Solutions to tasks for independent work:
Task 1. ((x)^(2))-8x+12=0
According to Vieta's theorem:
As usual, we start the selection with the piece:
Not suitable because the amount;
: the amount is just what you need.
Answer: ; .
Task 2.
And again our favorite Vieta theorem: the sum must be equal, and the product must be equal.
But since it must be not, but, we change the signs of the roots: and (in total).
Answer: ; .
Task 3.
Hmm... Where is that?
You need to move all the terms into one part:
The sum of the roots is equal to the product.
Okay, stop! The equation is not given. But Vieta's theorem is applicable only in the given equations. So first you need to give an equation. If you can’t lead, give up this idea and solve it in another way (for example, through a discriminant). Let me remind you that to give a quadratic equation means to make the leading coefficient equal:
Great. Then the sum of the roots is equal to and the product.
Here it’s as easy as shelling pears to choose: after all, it’s a prime number (sorry for the tautology).
Answer: ; .
Task 4.
The free member is negative. What's special about this? And the fact is that the roots will have different signs. And now, during the selection, we check not the sum of the roots, but the difference in their modules: this difference is equal, but a product.
So, the roots are equal to and, but one of them is minus. Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is. This means that the smaller root will have a minus: and, since.
Answer: ; .
Task 5.
What should you do first? That's right, give the equation:
Again: we select the factors of the number, and their difference should be equal to:
The roots are equal to and, but one of them is minus. Which? Their sum should be equal, which means that the minus will have a larger root.
Answer: ; .
Let me summarize:
- Vieta's theorem is used only in the quadratic equations given.
- Using Vieta's theorem, you can find the roots by selection, orally.
- If the equation is not given or no suitable pair of factors of the free term is found, then there are no whole roots, and you need to solve it in another way (for example, through a discriminant).
3. Method for selecting a complete square
If all terms containing the unknown are represented in the form of terms from abbreviated multiplication formulas - the square of the sum or difference - then after replacing variables, the equation can be presented in the form of an incomplete quadratic equation of the type.
For example:
Example 1:
Solve the equation: .
Solution:
Answer:
Example 2:
Solve the equation: .
Solution:
Answer:
In general, the transformation will look like this:
This implies: .
Doesn't remind you of anything? This is a discriminatory thing! That's exactly how we got the discriminant formula.
QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN THINGS
Quadratic equation- this is an equation of the form, where - the unknown, - the coefficients of the quadratic equation, - the free term.
Complete quadratic equation- an equation in which the coefficients are not equal to zero.
Reduced quadratic equation- an equation in which the coefficient, that is: .
Incomplete quadratic equation- an equation in which the coefficient and or the free term c are equal to zero:
- if the coefficient, the equation looks like: ,
- if there is a free term, the equation has the form: ,
- if and, the equation looks like: .
1. Algorithm for solving incomplete quadratic equations
1.1. Incomplete quadratic equation of the form, where, :
1) Let's express the unknown: ,
2) Check the sign of the expression:
- if, then the equation has no solutions,
- if, then the equation has two roots.
1.2. Incomplete quadratic equation of the form, where, :
1) Let’s take the common factor out of brackets: ,
2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:
1.3. Incomplete quadratic equation of the form, where:
This equation always has only one root: .
2. Algorithm for solving complete quadratic equations of the form where
2.1. Solution using discriminant
1) Let's bring the equation to standard form: ,
2) Let's calculate the discriminant using the formula: , which indicates the number of roots of the equation:
3) Find the roots of the equation:
- if, then the equation has roots, which are found by the formula:
- if, then the equation has a root, which is found by the formula:
- if, then the equation has no roots.
2.2. Solution using Vieta's theorem
The sum of the roots of the reduced quadratic equation (equation of the form where) is equal, and the product of the roots is equal, i.e. , A.
2.3. Solution by the method of selecting a complete square
If a quadratic equation of the form has roots, then it can be written in the form: .
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!
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