Square ur. Quadratic equations. Solving complete quadratic equations. Reduced and unreduced quadratic equations

I hope, after studying this article, you will learn how to find the roots of a complete quadratic equation.

With the help of the discriminant, only complete quadratic equations are solved; other methods are used to solve incomplete quadratic equations, which you will find in the article "Solving incomplete quadratic equations".

What quadratic equations are called complete? it equations of the form ax 2 + b x + c \u003d 0where the coefficients a, b and c are not equal to zero. So, to solve the full quadratic equation, you need to calculate the discriminant D.

D \u003d b 2 - 4ac.

Depending on what value the discriminant has, we will write down the answer.

If the discriminant is negative (D< 0),то корней нет.

If the discriminant is zero, then x \u003d (-b) / 2a. When the discriminant is a positive number (D\u003e 0),

then x 1 \u003d (-b - √D) / 2a, and x 2 \u003d (-b + √D) / 2a.

For example. Solve the equation x 2 - 4x + 4 \u003d 0.

D \u003d 4 2 - 4 4 \u003d 0

x \u003d (- (-4)) / 2 \u003d 2

Answer: 2.

Solve Equation 2 x 2 + x + 3 \u003d 0.

D \u003d 1 2 - 4 2 3 \u003d - 23

Answer: no roots.

Solve Equation 2 x 2 + 5x - 7 \u003d 0.

D \u003d 5 2 - 4 · 2 · (–7) \u003d 81

x 1 \u003d (-5 - √81) / (2 2) \u003d (-5 - 9) / 4 \u003d - 3.5

x 2 \u003d (-5 + √81) / (2 2) \u003d (-5 + 9) / 4 \u003d 1

Answer: - 3.5; 1.

So, we will present the solution of complete quadratic equations by the circuit in Figure 1.

Any complete quadratic equation can be solved using these formulas. You just need to be careful to ensure that the equation was written as a standard polynomial

and x 2 + bx + c, otherwise, you can make a mistake. For example, in writing the equation x + 3 + 2x 2 \u003d 0, you can erroneously decide that

a \u003d 1, b \u003d 3 and c \u003d 2. Then

D \u003d 3 2 - 4 · 1 · 2 \u003d 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 (in the first place should be the monomial with the largest exponent, that is and x 2 , then with less – bxand then a free member from.

When solving a reduced quadratic equation and a quadratic equation with an even coefficient at the second term, other formulas can also be used. Let's get to know these formulas as well. If in the full quadratic equation with the second term the coefficient is even (b \u003d 2k), then the equation can be solved using the formulas shown in the diagram in Figure 2.

A complete quadratic equation is called reduced if the coefficient at x 2 is equal to one and the equation takes the form x 2 + px + q \u003d 0... Such an equation can be given for the solution, or it is obtained by dividing all the coefficients of the equation by the coefficient andstanding at x 2 .

Figure 3 shows a scheme 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

3x 2 + 6x - 6 \u003d 0.

Let's solve this equation using the formulas shown in the diagram in Figure 1.

D \u003d 6 2 - 4 3 (- 6) \u003d 36 + 72 \u003d 108

√D \u003d √108 \u003d √ (363) \u003d 6√3

x 1 \u003d (-6 - 6√3) / (2 3) \u003d (6 (-1- √ (3))) / 6 \u003d –1 - √3

x 2 \u003d (-6 + 6√3) / (2 3) \u003d (6 (-1+ √ (3))) / 6 \u003d –1 + √3

Answer: -1 - √3; –1 + √3

It can be noted that the coefficient at x in this equation is an even number, that is, b \u003d 6 or b \u003d 2k, whence k \u003d 3. Then we will try to solve the equation according to the formulas shown in the diagram of the figure D 1 \u003d 3 2 - 3 · (- 6 ) \u003d 9 + 18 \u003d 27

√ (D 1) \u003d √27 \u003d √ (9 3) \u003d 3√3

x 1 \u003d (-3 - 3√3) / 3 \u003d (3 (-1 - √ (3))) / 3 \u003d - 1 - √3

x 2 \u003d (-3 + 3√3) / 3 \u003d (3 (-1 + √ (3))) / 3 \u003d - 1 + √3

Answer: -1 - √3; –1 + √3... Noticing that all the coefficients in this quadratic equation are divided by 3 and performing the division, we obtain the reduced quadratic equation x 2 + 2x - 2 \u003d 0 Solve this equation using the formulas for the reduced quadratic
equation figure 3.

D 2 \u003d 2 2 - 4 (- 2) \u003d 4 + 8 \u003d 12

√ (D 2) \u003d √12 \u003d √ (4 3) \u003d 2√3

x 1 \u003d (-2 - 2√3) / 2 \u003d (2 (-1 - √ (3))) / 2 \u003d - 1 - √3

x 2 \u003d (-2 + 2√3) / 2 \u003d (2 (-1+ √ (3))) / 2 \u003d - 1 + √3

Answer: -1 - √3; –1 + √3.

As you can see, when solving this equation using different formulas, we received the same answer. Therefore, having well mastered the formulas shown in the diagram in Figure 1, you can always solve any complete quadratic equation.

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We continue to study the topic “ solving equations". We have already met with linear equations and move on to get acquainted with quadratic equations.

First, we will analyze what a quadratic equation is, how it is written in general form, and give related definitions. After that, using examples, we will analyze in detail how incomplete quadratic equations are solved. Then we move on to solving the complete equations, obtain the formula for the roots, get acquainted with the discriminant of the quadratic equation and consider the solutions of typical examples. Finally, let's trace the relationship between roots and coefficients.

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What is a Quadratic Equation? Their types

First you need to clearly understand what a quadratic equation is. Therefore, it is logical to start talking about quadratic equations with the definition of a quadratic equation, as well as the definitions associated with it. After that, you can consider the main types of quadratic equations: reduced and non-reduced, as well as complete and incomplete equations.

Definition and examples of quadratic equations

Definition.

Quadratic equation Is an equation of the form a x 2 + b x + c \u003d 0 , where x is a variable, a, b and c are some numbers, and a is nonzero.

Let's say right away that quadratic equations are often called equations of the second degree. This is because the quadratic equation is algebraic equation second degree.

The sounded definition allows us to give examples of quadratic equations. So 2 x 2 + 6 x + 1 \u003d 0, 0.2 x 2 + 2.5 x + 0.03 \u003d 0, etc. Are quadratic equations.

Definition.

Numbers a, b and c are called coefficients of the quadratic equation a x 2 + b x + c \u003d 0, and the coefficient a is called the first, or the highest, or the coefficient at x 2, b is the second coefficient, or the coefficient at x, and c is the free term.

For example, let's take a quadratic equation of the form 5 x 2 −2 x − 3 \u003d 0, here the leading coefficient is 5, the second coefficient is −2, and the intercept is −3. Note that when the coefficients b and / or c are negative, as in the example just given, then the short form of the quadratic equation is 5 x 2 −2 x − 3 \u003d 0, not 5 x 2 + (- 2 ) X + (- 3) \u003d 0.

It should be noted that when the coefficients a and / or b are equal to 1 or −1, then they are usually not explicitly present in the quadratic equation, which is due to the peculiarities of writing such. For example, in a quadratic equation y 2 −y + 3 \u003d 0, the leading coefficient is one, and the coefficient at y is −1.

Reduced and unreduced quadratic equations

Reduced and non-reduced quadratic equations are distinguished depending on the value of the leading coefficient. Let us give the corresponding definitions.

Definition.

A quadratic equation in which the leading coefficient is 1 is called reduced quadratic equation... Otherwise the quadratic equation is unreduced.

According to this definition, quadratic equations x 2 −3 x + 1 \u003d 0, x 2 −x − 2/3 \u003d 0, etc. - given, in each of them the first coefficient is equal to one. And 5 x 2 −x − 1 \u003d 0, etc. - unreduced quadratic equations, their leading coefficients are different from 1.

From any non-reduced quadratic equation by dividing its both parts by the leading coefficient, you can go to the reduced one. This action is an equivalent transformation, that is, the reduced quadratic equation obtained in this way has the same roots as the original unreduced quadratic equation, or, like it, has no roots.

Let us analyze by example how the transition from an unreduced quadratic equation to a reduced one is performed.

Example.

From the equation 3 x 2 + 12 x − 7 \u003d 0, go to the corresponding reduced quadratic equation.

Decision.

It is enough for us to perform division of both sides of the original equation by the leading coefficient 3, it is nonzero, so we can perform this action. We have (3 x 2 + 12 x − 7): 3 \u003d 0: 3, which is the same, (3 x 2): 3+ (12 x): 3−7: 3 \u003d 0, and beyond (3: 3) x 2 + (12: 3) x − 7: 3 \u003d 0, whence. So we got the reduced quadratic equation, which is equivalent to the original one.

Answer:

Complete and incomplete quadratic equations

The definition of a quadratic equation contains the condition a ≠ 0. This condition is necessary for the equation a x 2 + b x + c \u003d 0 to be exactly quadratic, since at a \u003d 0 it actually becomes a linear equation of the form b x + c \u003d 0.

As for the coefficients b and c, they can be equal to zero, both separately and together. In these cases, the quadratic equation is called incomplete.

Definition.

The quadratic equation a x 2 + b x + c \u003d 0 is called incompleteif at least one of the coefficients b, c is equal to zero.

In its turn

Definition.

Full quadratic equation Is an equation in which all coefficients are nonzero.

These names are not given by chance. This will become clear from the following considerations.

If the coefficient b is equal to zero, then the quadratic equation takes the form a x 2 + 0 x + c \u003d 0, and it is equivalent to the equation a x 2 + c \u003d 0. If c \u003d 0, that is, the quadratic equation has the form a x 2 + b x + 0 \u003d 0, then it can be rewritten as a x 2 + b x \u003d 0. And with b \u003d 0 and c \u003d 0, we get the quadratic equation a · x 2 \u003d 0. The resulting equations differ from the full quadratic equation in that their left-hand sides do not contain either a term with variable x, or a free term, or both. Hence their name - incomplete quadratic equations.

So the equations x 2 + x + 1 \u003d 0 and −2 x 2 −5 x + 0.2 \u003d 0 are examples of complete quadratic equations, and x 2 \u003d 0, −2 x 2 \u003d 0.5 x 2 + 3 \u003d 0, −x 2 −5 · x \u003d 0 are incomplete quadratic equations.

Solving incomplete quadratic equations

From the information in the previous paragraph it follows that there is three kinds of incomplete quadratic equations:

• a x 2 \u003d 0, the coefficients b \u003d 0 and c \u003d 0 correspond to it;
• a x 2 + c \u003d 0 when b \u003d 0;
• and a x 2 + b x \u003d 0 when c \u003d 0.

Let us analyze in order how incomplete quadratic equations of each of these types are solved.

a x 2 \u003d 0

Let's start by solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a · x 2 \u003d 0. The equation a · x 2 \u003d 0 is equivalent to the equation x 2 \u003d 0, which is obtained from the original by dividing its both parts by a nonzero number a. Obviously, the root of the equation x 2 \u003d 0 is zero, since 0 2 \u003d 0. This equation has no other roots, which is explained, indeed, for any nonzero number p, the inequality p 2\u003e 0 holds, whence it follows that for p ≠ 0 the equality p 2 \u003d 0 is never achieved.

So, the incomplete quadratic equation a · x 2 \u003d 0 has a single root x \u003d 0.

As an example, let us give the solution to the incomplete quadratic equation −4 · x 2 \u003d 0. Equation x 2 \u003d 0 is equivalent to it, its only root is x \u003d 0, therefore, the original equation also has a unique root zero.

A short solution in this case can be formulated as follows:
−4 x 2 \u003d 0,
x 2 \u003d 0,
x \u003d 0.

a x 2 + c \u003d 0

Now we will consider how incomplete quadratic equations are solved, in which the coefficient b is zero, and c ≠ 0, that is, equations of the form a · x 2 + c \u003d 0. We know that the transfer of a term from one side of the equation to another with the opposite sign, as well as dividing both sides of the equation by a nonzero number, give an equivalent equation. Therefore, we can carry out the following equivalent transformations of the incomplete quadratic equation a x 2 + c \u003d 0:

• move c to the right side, which gives the equationax 2 \u003d −c,
• and divide both parts by a, we get.

The resulting equation allows us to draw conclusions about its roots. Depending on the values \u200b\u200bof a and c, the value of the expression can be negative (for example, if a \u003d 1 and c \u003d 2, then) or positive, (for example, if a \u003d −2 and c \u003d 6, then), it is not equal to zero , since by condition c ≠ 0. Let us examine separately the cases and.

If, then the equation has no roots. This statement follows from the fact that the square of any number is a non-negative number. It follows from this that when, then for any number p the equality cannot be true.

If, then the situation with the roots of the equation is different. In this case, if you remember about, then the root of the equation immediately becomes obvious, it is a number, since. It is easy to guess that the number is also the root of the equation, indeed,. This equation has no other roots, which can be shown, for example, by contradiction. Let's do it.

Let's denote the roots of the equation just voiced as x 1 and −x 1. Suppose the equation has one more root x 2 different from the indicated roots x 1 and −x 1. It is known that substitution of its roots in the equation instead of x turns the equation into a true numerical equality. For x 1 and −x 1 we have, and for x 2 we have. The properties of numerical equalities allow us to perform term-by-term subtraction of true numerical equalities, so subtracting the corresponding parts of the equalities gives x 1 2 - x 2 2 \u003d 0. The properties of actions with numbers allow you to rewrite the resulting equality as (x 1 - x 2) · (x 1 + x 2) \u003d 0. We know that the product of two numbers is equal to zero if and only if at least one of them is equal to zero. Therefore, it follows from the obtained equality that x 1 - x 2 \u003d 0 and / or x 1 + x 2 \u003d 0, which is the same, x 2 \u003d x 1 and / or x 2 \u003d −x 1. This is how we came to a contradiction, since at the beginning we said that the root of the equation x 2 is different from x 1 and −x 1. This proves that the equation has no roots other than and.

Let's summarize the information of this item. The incomplete quadratic equation a x 2 + c \u003d 0 is equivalent to the equation that

• has no roots if,
• has two roots and, if.

Consider examples of solving incomplete quadratic equations of the form a · x 2 + c \u003d 0.

Let's start with the quadratic equation 9 x 2 + 7 \u003d 0. After transferring the free term to the right side of the equation, it will take the form 9 · x 2 \u003d −7. Dividing both sides of the resulting equation by 9, we arrive at. Since there is a negative number on the right side, this equation has no roots, therefore, the original incomplete quadratic equation 9 · x 2 + 7 \u003d 0 has no roots.

Solve another incomplete quadratic equation −x 2 + 9 \u003d 0. Move the nine to the right: −x 2 \u003d −9. Now we divide both sides by −1, we get x 2 \u003d 9. On the right side there is a positive number, from which we conclude that or. Then we write down the final answer: the incomplete quadratic equation −x 2 + 9 \u003d 0 has two roots x \u003d 3 or x \u003d −3.

a x 2 + b x \u003d 0

It remains to deal with the solution of the last type of incomplete quadratic equations for c \u003d 0. Incomplete quadratic equations of the form a x 2 + b x \u003d 0 allows you to solve factorization method... Obviously, we can, located on the left side of the equation, for which it is enough to factor out the common factor x. This allows us to pass from the original incomplete quadratic equation to an equivalent equation of the form x · (a · x + b) \u003d 0. And this equation is equivalent to a combination of two equations x \u003d 0 and a x + b \u003d 0, the last of which is linear and has a root x \u003d −b / a.

So, the incomplete quadratic equation a x 2 + b x \u003d 0 has two roots x \u003d 0 and x \u003d −b / a.

To consolidate the material, we will analyze the solution of a specific example.

Example.

Solve the equation.

Decision.

Moving x out of parentheses gives the equation. It is equivalent to two equations x \u003d 0 and. We solve the received linear equation:, and after dividing the mixed number by an ordinary fraction, we find. Therefore, the roots of the original equation are x \u003d 0 and.

After getting the necessary practice, the solutions to such equations can be written briefly:

Answer:

x \u003d 0,.

Discriminant, the formula for the roots of a quadratic equation

There is a root formula for solving quadratic equations. Let's write quadratic formula:, where D \u003d b 2 −4 a c - so-called quadratic discriminant... The notation essentially means that.

It is useful to know how the root formula was obtained, and how it is applied when finding the roots of quadratic equations. Let's figure it out.

Derivation of the formula for the roots of a quadratic equation

Suppose we need to solve the quadratic equation a x 2 + b x + c \u003d 0. Let's perform some equivalent transformations:

• We can divide both sides of this equation by a nonzero number a, as a result we get the reduced quadratic equation.
• Now select a complete square on its left side:. After that, the equation will take the form.
• At this stage, it is possible to carry out the transfer of the last two terms to the right-hand side with the opposite sign, we have.
• And we also transform the expression on the right side:.

As a result, we come to an equation that is equivalent to the original quadratic equation a x 2 + b x + c \u003d 0.

We have already solved equations similar in form in the previous paragraphs, when we analyzed them. This allows us to draw the following conclusions regarding the roots of the equation:

• if, then the equation has no real solutions;
• if, then the equation has the form, therefore, whence its only root is visible;
• if, then or, which is the same or, that is, the equation has two roots.

Thus, the presence or absence of the roots of the equation, and hence the original quadratic equation, depends on the sign of the expression on the right side. In turn, the sign of this expression is determined by the sign of the numerator, since the denominator 4 · a 2 is always positive, that is, the sign of the expression b 2 −4 · a · c. This expression b 2 −4 a c was called the discriminant of the quadratic equation and marked with the letter D... From this, the essence of the discriminant is clear - by its value and sign, it is concluded whether the quadratic equation has real roots, and if so, what is their number - one or two.

Returning to the equation, we rewrite it using the discriminant notation:. And we draw conclusions:

• if D<0 , то это уравнение не имеет действительных корней;
• if D \u003d 0, then this equation has a single root;
• finally, if D\u003e 0, then the equation has two roots or, which, by virtue of it, can be rewritten in the form or, and after expanding and reducing the fractions to a common denominator, we obtain.

So we derived formulas for the roots of a quadratic equation, they have the form, where the discriminant D is calculated by the formula D \u003d b 2 −4 · a · c.

With their help, with a positive discriminant, you can calculate both real roots of the quadratic equation. When the discriminant is equal to zero, both formulas give the same root value corresponding to a unique solution to the quadratic equation. And with a negative discriminant, when trying to use the formula for the roots of a quadratic equation, we are faced with the extraction square root from a negative number, which takes us beyond and school curriculum... With negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found using the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

In practice, when solving quadratic equations, you can immediately use the root formula, with which you can calculate their values. But this is more about finding complex roots.

However, in the school algebra course, it is usually not about complex, but about real roots of a quadratic equation. In this case, it is advisable to first find the discriminant before using the formulas for the roots of the quadratic equation, make sure that it is non-negative (otherwise, we can conclude that the equation has no real roots), and only after that calculate the values \u200b\u200bof the roots.

The above reasoning allows us to write quadratic equation solver... To solve the quadratic equation a x 2 + b x + c \u003d 0, you need:

• by the discriminant formula D \u003d b 2 −4 · a · c calculate its value;
• conclude that the quadratic equation has no real roots if the discriminant is negative;
• calculate the only root of the equation by the formula if D \u003d 0;
• find two real roots of a quadratic equation using the root formula if the discriminant is positive.

Here we just note that if the discriminant is equal to zero, the formula can also be used, it will give the same value as.

You can proceed to examples of using the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Consider solutions to three quadratic equations with positive, negative and zero discriminants. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's start.

Example.

Find the roots of the equation x 2 + 2 x − 6 \u003d 0.

Decision.

In this case, we have the following coefficients of the quadratic equation: a \u003d 1, b \u003d 2 and c \u003d −6. According to the algorithm, first you need to calculate the discriminant, for this we substitute the indicated a, b and c into the discriminant formula, we have D \u003d b 2 −4 a c \u003d 2 2 −4 1 (−6) \u003d 4 + 24 \u003d 28... Since 28\u003e 0, that is, the discriminant is greater than zero, the quadratic equation has two real roots. We find them by the root formula, we get, here you can simplify the expressions obtained by doing factoring out the sign of the root with the subsequent reduction of the fraction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4x2 + 28x − 49 \u003d 0.

Decision.

We start by finding the discriminant: D \u003d 28 2 −4 (−4) (−49) \u003d 784−784 \u003d 0... Therefore, this quadratic equation has a single root, which we find as, that is,

Answer:

x \u003d 3.5.

It remains to consider the solution of quadratic equations with negative discriminant.

Example.

Solve the equation 5 y 2 + 6 y + 2 \u003d 0.

Decision.

Here are the coefficients of the quadratic equation: a \u003d 5, b \u003d 6 and c \u003d 2. Substituting these values \u200b\u200binto the discriminant formula, we have D \u003d b 2 −4 a c \u003d 6 2 −4 5 2 \u003d 36−40 \u003d −4... The discriminant is negative, therefore, this quadratic equation has no real roots.

If you need to indicate complex roots, then we apply the well-known formula for the roots of the quadratic equation, and perform complex number operations:

Answer:

there are no real roots, complex roots are as follows:.

Once again, we note that if the discriminant of the quadratic equation is negative, then at school they usually immediately write down an answer in which they indicate that there are no real roots, and complex roots are not found.

Root formula for even second coefficients

The formula for the roots of a quadratic equation, where D \u003d b 2 −4 a c, makes it possible to obtain a formula of a more compact form that allows solving quadratic equations with an even coefficient at x (or simply with a coefficient of the form 2 n, for example, or 14 ln5 \u003d 2 7 ln5). Let's take it out.

Let's say we need to solve a quadratic equation of the form a x 2 + 2 n x + c \u003d 0. Let's find its roots using the formula we know. To do this, calculate the discriminant D \u003d (2 n) 2 −4 a c \u003d 4 n 2 −4 a c \u003d 4 (n 2 −a c), and then use the root formula:

Let us denote the expression n 2 −a · c as D 1 (sometimes it is denoted by D "). Then the formula for the roots of the considered quadratic equation with the second coefficient 2 n takes the form , where D 1 \u003d n 2 - a · c.

It is easy to see that D \u003d 4 · D 1, or D 1 \u003d D / 4. In other words, D 1 is the fourth part of the discriminant. It is clear that the sign of D 1 is the same as the sign of D. That is, the sign of D 1 is also an indicator of the presence or absence of the roots of a quadratic equation.

So, to solve the quadratic equation with the second coefficient 2 n, you need

• Calculate D 1 \u003d n 2 −a · c;
• If D 1<0 , то сделать вывод, что действительных корней нет;
• If D 1 \u003d 0, then calculate the only root of the equation by the formula;
• If D 1\u003e 0, then find two real roots by the formula.

Let's consider the solution of an example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5x2 −6x − 32 \u003d 0.

Decision.

The second coefficient of this equation can be represented as 2 · (−3). That is, you can rewrite the original quadratic equation in the form 5 x 2 + 2 (−3) x − 32 \u003d 0, here a \u003d 5, n \u003d −3 and c \u003d −32, and calculate the fourth part of the discriminant: D 1 \u003d n 2 −a c \u003d (- 3) 2 −5 (−32) \u003d 9 + 160 \u003d 169... Since its value is positive, the equation has two real roots. Let's find them using the corresponding root formula:

Note that it was possible to use the usual formula for the roots of a quadratic equation, but in this case, more computational work would have to be done.

Answer:

Simplifying Quadratic Equations

Sometimes, before embarking on the calculation of the roots of a quadratic equation by formulas, it does not hurt to ask the question: "Is it possible to simplify the form of this equation"? Agree that in terms of calculations it will be easier to solve the quadratic equation 11 x 2 −4 x − 6 \u003d 0 than 1100 x 2 −400 x − 600 \u003d 0.

Usually, a simplification of the form of a quadratic equation is achieved by multiplying or dividing both parts of it by some number. For example, in the previous paragraph, we managed to simplify the equation 1100x2 −400x − 600 \u003d 0 by dividing both sides by 100.

A similar transformation is carried out with quadratic equations, the coefficients of which are not. In this case, both sides of the equation are usually divided by absolute values its coefficients. For example, let's take the quadratic equation 12 x 2 −42 x + 48 \u003d 0. the absolute values \u200b\u200bof its coefficients: GCD (12, 42, 48) \u003d GCD (GCD (12, 42), 48) \u003d GCD (6, 48) \u003d 6. Dividing both sides of the original quadratic equation by 6, we arrive at the equivalent quadratic equation 2 x 2 −7 x + 8 \u003d 0.

And multiplication of both sides of a quadratic equation is usually done to get rid of fractional coefficients. In this case, the multiplication is carried out by the denominators of its coefficients. For example, if both sides of a quadratic equation are multiplied by the LCM (6, 3, 1) \u003d 6, then it will take on a simpler form x 2 + 4 x − 18 \u003d 0.

In conclusion of this paragraph, we note that we almost always get rid of the minus at the leading coefficient of the quadratic equation, changing the signs of all terms, which corresponds to multiplying (or dividing) both parts by −1. For example, usually from the quadratic equation −2x2 −3x + 7 \u003d 0 one goes over to the solution 2x2 + 3x − 7 \u003d 0.

Relationship between roots and coefficients of a quadratic equation

The formula for the roots of a quadratic equation expresses the roots of an equation in terms of its coefficients. Based on the root formula, you can get other dependencies between the roots and the coefficients.

The most famous and applicable formulas are from Vieta's theorem of the form and. In particular, for the given quadratic equation, the sum of the roots is equal to the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by the form of the quadratic equation 3 x 2 −7 x + 22 \u003d 0, we can immediately say that the sum of its roots is 7/3, and the product of the roots is 22/3.

Using the already written formulas, you can get a number of other relationships between the roots and the coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation through its coefficients:.

List of references.

• Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
• A. G. Mordkovich Algebra. 8th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., Erased. - M .: Mnemozina, 2009 .-- 215 p .: ill. ISBN 978-5-346-01155-2.

Just. By formulas and clear, simple rules. At the first stage

it is necessary to reduce the given equation to standard view, i.e. to look:

If the equation is already given to you in this form, you do not need to do the first step. The most important thing is right

determine all the coefficients, and, b and c.

Formula for finding the roots of a quadratic equation.

An expression under the root sign is called discriminant ... As you can see, to find x, we

use only a, b and c. Those. coefficients from quadratic equation... Just carefully substitute

meaning a, b and c into this formula and count. Substitute with by their signs!

for example, in the equation:

and =1; b = 3; c = -4.

Substitute the values \u200b\u200band write:

The example is almost solved:

This is the answer.

The most common mistakes are confusion with meaning signs. a, band from... Rather, with substitution

negative values \u200b\u200binto the formula for calculating the roots. Here a detailed notation of the formula saves

with specific numbers. If you have computational problems, do it!

Suppose you need to solve this example:

Here a = -6; b = -5; c = -1

We paint everything in detail, carefully, without missing anything with all the signs and brackets:

Quadratic equations often look slightly different. For example, like this:

Now, take note of the best practices that will dramatically reduce errors.

First reception... Don't be lazy before solution of the quadratic equation bring it to standard form.

What does this mean?

Let's say, after any transformations, you got the following equation:

Don't rush to write the root formula! You will almost certainly mix up the odds. a, b and c.

Build the example correctly. First, the X is squared, then without the square, then the free member. Like this:

Get rid of the minus. How? You have to multiply the whole equation by -1. We get:

But now you can safely write down the formula for the roots, calculate the discriminant and complete the example.

Do it yourself. You should have roots 2 and -1.

Reception second. Check the roots! By vieta's theorem.

To solve the given quadratic equations, i.e. if the coefficient

x 2 + bx + c \u003d 0,

then x 1 x 2 \u003d c

x 1 + x 2 \u003d -b

For a complete quadratic equation in which a ≠ 1:

x 2 +bx +c=0,

divide the whole equation by and:

→ →

where x 1 and x 2 - the roots of the equation.

Reception third... If your equation contains fractional coefficients, get rid of fractions! Multiply

common denominator equation.

Output. Practical advice:

1. Before solving, we bring the quadratic equation to the standard form, build it right.

2. If there is a negative coefficient in front of the x in the square, we eliminate it by multiplying the total

equations 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, the coefficient is equal to one, the solution can be easily checked by

Quadratic equations... General information.

IN quadratic x must be present in the square (that's why it is called

"Square"). In addition to him, the equation may (or may not be!) Just x (in the first degree) and

just a number (free member). And there should be no x's to a degree greater than two.

General algebraic equation.

where x - free variable, a, b, c - coefficients, and a≠0 .

for example:

Expression called square trinomial.

The elements of the quadratic equation have their own names:

Called the first or highest coefficient,

Called the second or coefficient at,

· Called a free member.

Complete quadratic equation.

These quadratic equations have a complete set of terms on the left. X squared with

coefficient and, x to the first power with a coefficient b and free member from. INall odds

must be nonzero.

Incomplete is called a quadratic equation in which at least one of the coefficients, except

the highest one (either the second coefficient or the free term) is equal to zero.

Let's pretend that b \u003d 0, - x disappears in the first degree. It turns out, for example:

2x 2 -6x \u003d 0,

Etc. And if both coefficients, b and c are equal to zero, then everything is even easier, eg:

2x 2 \u003d 0,

Note that the x squared is present in all equations.

Why and can't be zero? Then the x squared disappears and the equation becomes linear .

And it is decided in a completely different way ...

An incomplete quadratic equation differs from classical (complete) equations in that its factors or intercept are equal to zero. The graph of such functions are parabolas. Depending on their general appearance, they are divided into 3 groups. The principles of solving for all types of equations are the same.

There is nothing difficult in determining the type of an incomplete polynomial. It is best to consider the main differences with illustrative examples:

1. If b \u003d 0, then the equation is ax 2 + c \u003d 0.
2. If c \u003d 0, then the expression ax 2 + bx \u003d 0 should be solved.
3. If b \u003d 0 and c \u003d 0, then the polynomial becomes an equality of the type ax 2 \u003d 0.

The latter case is more of a theoretical possibility and never occurs in knowledge testing tasks, since the only valid value of the variable x in the expression is zero. In the future, methods and examples of solving incomplete quadratic equations 1) and 2) types will be considered.

General algorithm for finding variables and examples with a solution

Regardless of the type of equation, the solution algorithm boils down to the following steps:

1. Bring the expression to a form convenient for finding roots.
2. Perform calculations.
3. Record your answer.

The easiest way to solve incomplete equations is by factoring the left side and leaving zero on the right. Thus, the formula for an incomplete quadratic equation for finding the roots is reduced to calculating the value of x for each of the factors.

You can only learn how to solve it in practice, so let's consider a specific example of finding the roots of an incomplete equation:

As you can see, in this case b \u003d 0. We factor the left side and get the expression:

4 (x - 0.5) ⋅ (x + 0.5) \u003d 0.

Obviously, the product is zero when at least one of the factors is zero. The values \u200b\u200bof the variable x1 \u003d 0.5 and (or) x2 \u003d -0.5 meet these requirements.

In order to easily and quickly cope with the problem of factoring a square trinomial into factors, you should remember the following formula:

If there is no free term in the expression, the task is greatly simplified. It will be enough just to find and take out the common denominator. For clarity, consider an example of how to solve incomplete quadratic equations of the form ax2 + bx \u003d 0.

Let's take the variable x out of the brackets and get the following expression:

x ⋅ (x + 3) \u003d 0.

Guided by logic, we come to the conclusion that x1 \u003d 0, and x2 \u003d -3.

Traditional solution and incomplete quadratic equations

What will happen if you apply the discriminant formula and try to find the roots of the polynomial, with the coefficients equal to zero? Let's take an example from a collection of typical tasks for the exam in mathematics in 2017, solve it using standard formulas and the method of factoring.

7x 2 - 3x \u003d 0.

Let's calculate the value of the discriminant: D \u003d (-3) 2 - 4 ⋅ (-7) ⋅ 0 \u003d 9. It turns out that the polynomial has two roots:

Now, let's solve the equation by factoring and compare the results.

X ⋅ (7x + 3) \u003d 0,

2) 7x + 3 \u003d 0,
7x \u003d -3,
x \u003d -.

As you can see, both methods give the same result, but solving the equation by the second method turned out to be much easier and faster.

Vieta's theorem

And what to do with the beloved Vieta theorem? Can this method be used with an incomplete trinomial? Let's try to understand the aspects of reducing incomplete equations to the classical form ax2 + bx + c \u003d 0.

In fact, it is possible to apply Vieta's theorem in this case. It is only necessary to bring the expression to a general form, replacing the missing members with zero.

For example, for b \u003d 0 and a \u003d 1, in order to eliminate the likelihood of confusion, the task should be written in the form: ax2 + 0 + c \u003d 0. Then the ratio of the sum and product of the roots and factors of the polynomial can be expressed as follows:

Theoretical calculations help to get acquainted with the essence of the issue, and always require practicing skills in solving specific problems. Let's turn again to the reference book of typical tasks for the exam and find a suitable example:

Let us write the expression in a form convenient for applying Vieta's theorem:

x 2 + 0 - 16 \u003d 0.

The next step is to create a system of conditions:

Obviously, the roots of a square polynomial will be x 1 \u003d 4 and x 2 \u003d -4.

Now, let's practice bringing the equation to a general form. Take the following example: 1/4 × x 2 - 1 \u003d 0

In order to apply Vieta's theorem to an expression, it is necessary to get rid of the fraction. Multiply the left and right sides by 4, and look at the result: x2– 4 \u003d 0. The resulting equality is ready to be solved by Vieta's theorem, but it is much easier and faster to get the answer simply by transferring c \u003d 4 to the right side of the equation: x2 \u003d 4.

Summing up, it should be said that the best way to solve incomplete equations is factorization, which is the simplest and fastest method. If you encounter difficulties in the process of finding roots, you can turn to the traditional method of finding roots through the discriminant.