The leading coefficient of the quadratic equation. Incomplete quadratic equations

Quadratic equation - easy to solve! *Further in the text "KU". Friends, it would seem that in mathematics it can be easier than solving such an equation. But something told me that many people have problems with him. I decided to see how many impressions Yandex gives per request per month. Here's what happened, take a look:

What does it mean? This means that about 70,000 people a month are looking for this information, what does this summer have to do with it, and what will happen among school year- requests will be twice as large. This is not surprising, because those guys and girls who have long graduated from school and are preparing for the exam are looking for this information, and schoolchildren are also trying to refresh their memory.

Despite the fact that there are a lot of sites that tell how to solve this equation, I decided to also contribute and publish the material. Firstly, I want visitors to come to my site on this request; secondly, in other articles, when the speech “KU” comes up, I will give a link to this article; thirdly, I will tell you a little more about his solution than is usually stated on other sites. Let's get started! The content of the article:

A quadratic equation is an equation of the form:

where coefficients a,band with arbitrary numbers, with a≠0.

In the school course, the material is given in the following form - the division of equations into three classes is conditionally done:

1. Have two roots.

2. * Have only one root.

3. Have no roots. It is worth noting here that they do not have real roots

How are roots calculated? Just!

We calculate the discriminant. Under this "terrible" word lies a very simple formula:

The root formulas are as follows:

*These formulas must be known by heart.

You can immediately write down and solve:

Example:

1. If D > 0, then the equation has two roots.

2. If D = 0, then the equation has one root.

3. If D< 0, то уравнение не имеет действительных корней.

Let's look at the equation:

By this occasion when the discriminant zero, the school course says that one root is obtained, here it is equal to nine. That's right, it is, but...

This representation is somewhat incorrect. In fact, there are two roots. Yes, yes, do not be surprised, it turns out two equal root, and to be mathematically precise, two roots should be written in the answer:

x 1 = 3 x 2 = 3

But this is so - a small digression. At school, you can write down and say that there is only one root.

Now the following example:

As we know, the root of a negative number is not extracted, so there is no solution in this case.

That's the whole decision process.

Quadratic function.

Here is how the solution looks geometrically. This is extremely important to understand (in the future, in one of the articles, we will analyze in detail the solution of a quadratic inequality).

This is a function of the form:

where x and y are variables

a, b, c - given numbers, where a ≠ 0

The graph is a parabola:

That is, it turns out that by solving a quadratic equation with "y" equal to zero, we find the points of intersection of the parabola with the x-axis. There can be two of these points (the discriminant is positive), one (the discriminant is zero) or none (the discriminant is negative). Details about quadratic function You can view article by Inna Feldman.

Consider examples:

Example 1: Decide 2x 2 +8 x–192=0

a=2 b=8 c= -192

D = b 2 –4ac = 8 2 –4∙2∙(–192) = 64+1536 = 1600

Answer: x 1 = 8 x 2 = -12

* You could immediately divide the left and right sides of the equation by 2, that is, simplify it. The calculations will be easier.

Example 2: Decide x2–22 x+121 = 0

a=1 b=-22 c=121

D = b 2 –4ac =(–22) 2 –4∙1∙121 = 484–484 = 0

We got that x 1 \u003d 11 and x 2 \u003d 11

In the answer, it is permissible to write x = 11.

Answer: x = 11

Example 3: Decide x 2 –8x+72 = 0

a=1 b= -8 c=72

D = b 2 –4ac =(–8) 2 –4∙1∙72 = 64–288 = –224

The discriminant is negative, there is no solution in real numbers.

Answer: no solution

The discriminant is negative. There is a solution!

Here we will talk about solving the equation in the case when a negative discriminant is obtained. Do you know anything about complex numbers? I will not go into detail here about why and where they arose and what their specific role and necessity in mathematics is, this is a topic for a large separate article.

The concept of a complex number.

A bit of theory.

A complex number z is a number of the form

z = a + bi

where a and b are real numbers, i is the so-called imaginary unit.

a+bi is a SINGLE NUMBER, not an addition.

The imaginary unit is equal to the root of minus one:

Now consider the equation:

Get two conjugate roots.

Incomplete quadratic equation.

Consider special cases, this is when the coefficient "b" or "c" is equal to zero (or both are equal to zero). They are solved easily without any discriminants.

Case 1. Coefficient b = 0.

The equation takes the form:

Let's transform:

Example:

4x 2 -16 = 0 => 4x 2 =16 => x 2 = 4 => x 1 = 2 x 2 = -2

Case 2. Coefficient c = 0.

The equation takes the form:

Transform, factorize:

*The product is equal to zero when at least one of the factors is equal to zero.

Example:

9x 2 –45x = 0 => 9x (x–5) =0 => x = 0 or x–5 =0

x 1 = 0 x 2 = 5

Case 3. Coefficients b = 0 and c = 0.

Here it is clear that the solution to the equation will always be x = 0.

Useful properties and patterns of coefficients.

There are properties that allow solving equations with large coefficients.

ax 2 + bx+ c=0 equality

a + b+ c = 0, then

— if for the coefficients of the equation ax 2 + bx+ c=0 equality

a+ with =b, then

These properties help solve a certain kind of equation.

Example 1: 5001 x 2 –4995 x – 6=0

The sum of the coefficients is 5001+( – 4995)+(– 6) = 0, so

Example 2: 2501 x 2 +2507 x+6=0

Equality a+ with =b, means

Regularities of coefficients.

1. If in the equation ax 2 + bx + c \u003d 0 the coefficient "b" is (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are

ax 2 + (a 2 +1) ∙ x + a \u003d 0 \u003d\u003e x 1 \u003d -a x 2 \u003d -1 / a.

Example. Consider the equation 6x 2 +37x+6 = 0.

x 1 \u003d -6 x 2 \u003d -1/6.

2. If in the equation ax 2 - bx + c \u003d 0, the coefficient "b" is (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are

ax 2 - (a 2 + 1) ∙ x + a \u003d 0 \u003d\u003e x 1 \u003d a x 2 \u003d 1 / a.

Example. Consider the equation 15x 2 –226x +15 = 0.

x 1 = 15 x 2 = 1/15.

3. If in the equation ax 2 + bx - c = 0 coefficient "b" equals (a 2 – 1), and the coefficient “c” numerically equal to the coefficient "a", then its roots are equal

ax 2 + (a 2 -1) ∙ x - a \u003d 0 \u003d\u003e x 1 \u003d - a x 2 \u003d 1 / a.

Example. Consider the equation 17x 2 + 288x - 17 = 0.

x 1 \u003d - 17 x 2 \u003d 1/17.

4. If in the equation ax 2 - bx - c \u003d 0, the coefficient "b" is equal to (a 2 - 1), and the coefficient c is numerically equal to the coefficient "a", then its roots are

ax 2 - (a 2 -1) ∙ x - a \u003d 0 \u003d\u003e x 1 \u003d a x 2 \u003d - 1 / a.

Example. Consider the equation 10x2 - 99x -10 = 0.

x 1 \u003d 10 x 2 \u003d - 1/10

Vieta's theorem.

Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, one can express the sum and product of the roots of an arbitrary KU in terms of its coefficients.

45 = 1∙45 45 = 3∙15 45 = 5∙9.

In sum, the number 14 gives only 5 and 9. These are the roots. With a certain skill, using the presented theorem, you can solve many quadratic equations immediately orally.

Vieta's theorem, moreover. convenient because after solving the quadratic equation in the usual way (through the discriminant), the resulting roots can be checked. I recommend doing this all the time.

TRANSFER METHOD

With this method, the coefficient "a" is multiplied by the free term, as if "transferred" to it, which is why it is called transfer method. This method is used when it is easy to find the roots of an equation using Vieta's theorem and, most importantly, when the discriminant is an exact square.

If a a± b+c≠ 0, then the transfer technique is used, for example:

2X 2 – 11x+ 5 = 0 (1) => X 2 – 11x+ 10 = 0 (2)

According to the Vieta theorem in equation (2), it is easy to determine that x 1 \u003d 10 x 2 \u003d 1

The obtained roots of the equation must be divided by 2 (since the two were “thrown” from x 2), we get

x 1 \u003d 5 x 2 \u003d 0.5.

What is the rationale? See what's happening.

The discriminants of equations (1) and (2) are:

If you look at the roots of the equations, then only different denominators are obtained, and the result depends precisely on the coefficient at x 2:

The second (modified) roots are 2 times larger.

Therefore, we divide the result by 2.

*If we roll three of a kind, then we divide the result by 3, and so on.

Answer: x 1 = 5 x 2 = 0.5

sq. ur-ie and the exam.

I will say briefly about its importance - YOU SHOULD BE ABLE TO DECIDE quickly and without thinking, you need to know the formulas of the roots and the discriminant by heart. A lot of the tasks that are part of the USE tasks come down to solving a quadratic equation (including geometric ones).

What is worth noting!

1. The form of the equation can be "implicit". For example, the following entry is possible:

15+ 9x 2 - 45x = 0 or 15x+42+9x 2 - 45x=0 or 15 -5x+10x 2 = 0.

You need to bring it to a standard form (so as not to get confused when solving).

2. Remember that x is an unknown value and it can be denoted by any other letter - t, q, p, h and others.

An incomplete quadratic equation differs from classical (complete) equations in that its factors or free term are equal to zero. The graph of such functions are parabolas. Depending on the general appearance, they are divided into 3 groups. The principles of solution 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 in illustrative examples:

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

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

General Algorithm for Finding Variables and Examples with a Solution

Regardless of the type of equation, the solution algorithm is reduced to the following steps:

Bring the expression to a form convenient for finding roots.
Make calculations.
Write down the answer.

It is easiest to solve incomplete equations by factoring the left side and leaving zero on the right side. 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 learn how to solve only in practice, so consider specific example finding the roots of an incomplete equation:

As you can see, in this case b = 0. We factorize the left side and get the expression:

4(x - 0.5) ⋅ (x + 0.5) = 0.

Obviously, the product is equal to zero when at least one of the factors is equal to zero. Similar requirements are met by the values of the variable x1 = 0.5 and (or) x2 = -0.5.

In order to easily and quickly cope with the task of decomposition square trinomial multipliers, 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 = 0.

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

x ⋅ (x + 3) = 0.

Based on logic, we conclude that x1 = 0 and x2 = -3.

The traditional way of solving and incomplete quadratic equations

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

7x 2 - 3x = 0.

Calculate the value of the discriminant: D = (-3)2 - 4 ⋅ (-7) ⋅ 0 = 9. It turns out that the polynomial has two roots:

Now, solve the equation by factoring and compare the results.

X ⋅ (7x + 3) = 0,

2) 7x + 3 = 0,
7x=-3,
x = -.

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

Vieta's theorem

But what to do with the beloved Vieta theorem? Can this method be applied with an incomplete trinomial? Let's try to understand the aspects of reducing incomplete equations to the classical form ax2 + bx + c = 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 terms with zero.

For example, with b = 0 and a = 1, in order to eliminate the possibility of confusion, the task should be written in the form: ax2 + 0 + c = 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 skill development when solving specific tasks. Let's turn again to the reference book of typical tasks for the exam and find a suitable example:

We write the expression in a form convenient for applying the Vieta theorem:

x2 + 0 - 16 = 0.

The next step is to create a system of conditions:

Obviously, the roots of the 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 = 0

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

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

A quadratic equation is an equation of the form a*x^2 +b*x+c=0, where a,b,c are some arbitrary real (real) numbers, and x is a variable. And the number a is not equal to 0.

The numbers a,b,c are called coefficients. The number a - is called the leading coefficient, the number b is the coefficient at x, and the number c is called the free member. Other names are also found in some literature. The number a is called the first coefficient, and the number b is called the second coefficient.

Classification of quadratic equations

Quadratic equations have their own classification.

By the presence of coefficients:

1. Full

2. Incomplete

By the value of the coefficient of the highest degree of the unknown(to the value of the leading coefficient):

1. Given

2. Not reduced

Quadratic equation called complete if it contains all three coefficients and they are nonzero. General form full quadratic equation: a*x^2 +b*x+c=0;

Quadratic equation called incomplete if in the equation a * x ^ 2 + b * x + c \u003d 0 one of the coefficients b or c is equal to zero (b \u003d 0 or c \u003d 0), however, an incomplete quadratic equation will also be an equation in which both coefficient b and coefficient c are simultaneously equal to zero (both b=0 and c=0).

It is worth noting that nothing is said here about the leading coefficient, since, by the definition of a quadratic equation, it must be different from zero.

given if its leading coefficient equal to one(a=1). General view of the given quadratic equation: x^2 +d*x+e=0.

The quadratic equation is called unreduced, if the leading coefficient in the equation is non-zero. General view of the unreduced quadratic equation: a*x^2 +b*x+c=0.

It should be noted that any non-reduced quadratic equation can be reduced to the reduced one. To do this, it is necessary to divide the coefficients of the quadratic equation by the leading coefficient.

Quadratic Examples

Consider an example: we have the equation 2*x^2 - 6*x+7 =0;

Let's transform it into the above equation. The leading coefficient is 2. Let's divide the coefficients of our equation by it and write down the answer.

x^2 - 3*x+3.5 =0;

As you noticed, on the right side of the quadratic equation is a polynomial of the second degree a * x ^ 2 + b * x + c. It is also called a square trinomial.

This topic may seem complicated at first due to the many not-so-simple formulas. Not only do the quadratic equations themselves have long entries, but the roots are also found through the discriminant. There are three new formulas in total. Not very easy to remember. This is possible only after the frequent solution of such equations. Then all the formulas will be remembered by themselves.

General view of the quadratic equation

Here their explicit notation is proposed, when the largest degree is written first, and then - in descending order. Often there are situations when the terms stand apart. Then it is better to rewrite the equation in descending order of the degree of the variable.

Let us introduce notation. They are presented in the table below.

If we accept these notations, all quadratic equations are reduced to the following notation.

Moreover, the coefficient a ≠ 0. Let this formula be denoted by number one.

When the equation is given, it is not clear how many roots will be in the answer. Because one of three options is always possible:

the solution will have two roots;
the answer will be one number;
The equation has no roots at all.

And while the decision is not brought to the end, it is difficult to understand which of the options will fall out in a particular case.

Types of records of quadratic equations

Tasks may have different entries. They don't always look like general formula quadratic equation. Sometimes it will lack some terms. What was written above is complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.

Moreover, only the terms for which the coefficients "b" and "c" can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of the equations will be as follows:

So, there are only two types, in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second number three.

The discriminant and the dependence of the number of roots on its value

This number must be known in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have the number four.

After substituting the values of the coefficients into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. With a negative number, the roots of the quadratic equation will be absent. If it is equal to zero, the answer will be one.

How is a complete quadratic equation solved?

In fact, consideration of this issue has already begun. Because first you need to find the discriminant. After it is clarified that there are roots of the quadratic equation, and their number is known, you need to use the formulas for the variables. If there are two roots, then you need to apply such a formula.

Since it contains the “±” sign, there will be two values. Signed expression square root is the discriminant. Therefore, the formula can be rewritten in a different way.

Formula five. From the same record it can be seen that if the discriminant is zero, then both roots will take the same values.

If the solution of quadratic equations has not yet been worked out, then it is better to write down the values of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.

How is an incomplete quadratic equation solved?

Everything is much simpler here. Even there is no need for additional formulas. And you won't need those that have already been written for the discriminant and the unknown.

First consider incomplete equation at number two. In this equality, it is supposed to take the unknown value out of the bracket and solve the linear equation, which will remain in the brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a factor consisting of the variable itself. The second is obtained by solving a linear equation.

The incomplete equation at number three is solved by transferring the number from the left side of the equation to the right. Then you need to divide by the coefficient in front of the unknown. It remains only to extract the square root and do not forget to write it down twice with opposite signs.

The following are some actions that help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings are the cause of poor grades when studying the extensive topic "Quadric Equations (Grade 8)". Subsequently, these actions will not need to be constantly performed. Because there will be a stable habit.

First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without the degree and the last - just a number.

If a minus appears before the coefficient "a", then it can complicate the work for a beginner to study quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by "-1". This means that all terms will change sign to the opposite.

In the same way, it is recommended to get rid of fractions. Simply multiply the equation by the appropriate factor so that the denominators cancel out.

Examples

It is required to solve the following quadratic equations:

x 2 - 7x \u003d 0;

15 - 2x - x 2 \u003d 0;

x 2 + 8 + 3x = 0;

12x + x 2 + 36 = 0;

(x+1) 2 + x + 1 = (x+1)(x+2).

The first equation: x 2 - 7x \u003d 0. It is incomplete, therefore it is solved as described for formula number two.

After bracketing, it turns out: x (x - 7) \u003d 0.

The first root takes the value: x 1 = 0. The second will be found from linear equation: x - 7 = 0. It is easy to see that x 2 = 7.

Second equation: 5x2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.

After transferring 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be numbers: x 1 = √6, x 2 = - √6.

Third equation: 15 - 2x - x 2 \u003d 0. Here and below, the solution of quadratic equations will begin by rewriting them in standard view: - x 2 - 2x + 15 = 0. Now it's time to use the second useful advice and multiply everything by minus one. It turns out x 2 + 2x - 15 \u003d 0. According to the fourth formula, you need to calculate the discriminant: D \u003d 2 2 - 4 * (- 15) \u003d 4 + 60 \u003d 64. It is a positive number. From what was said above, it turns out that the equation has two roots. They need to be calculated according to the fifth formula. According to it, it turns out that x \u003d (-2 ± √64) / 2 \u003d (-2 ± 8) / 2. Then x 1 \u003d 3, x 2 \u003d - 5.

The fourth equation x 2 + 8 + 3x \u003d 0 is converted to this: x 2 + 3x + 8 \u003d 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: "There are no roots."

The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x \u003d -12 / (2 * 1) \u003d -6.

The sixth equation (x + 1) 2 + x + 1 = (x + 1) (x + 2) requires transformations, which consist in the fact that you need to bring like terms, before opening the brackets. In place of the first one there will be such an expression: x 2 + 2x + 1. After equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x \u003d 0. It has become incomplete . Similar to it has already been considered a little higher. The roots of this will be the numbers 0 and 1.

5x (x - 4) = 0

5 x = 0 or x - 4 = 0

x = ± √ 25/4

Having learned to solve equations of the first degree, of course, I want to work with others, in particular, with equations of the second degree, which are otherwise called quadratic.

Quadratic equations are equations of the type ax² + bx + c = 0, where the variable is x, the numbers will be - a, b, c, where a is not equal to zero.

If in a quadratic equation one or the other coefficient (c or b) is equal to zero, then this equation will refer to an incomplete quadratic equation.

How to solve an incomplete quadratic equation if students have only been able to solve equations of the first degree so far? Consider incomplete quadratic equations different types and simple ways their decisions.

a) If the coefficient c is equal to 0, and the coefficient b is not equal to zero, then ax ² + bx + 0 = 0 is reduced to an equation of the form ax ² + bx = 0.

To solve such an equation, you need to know the formula for solving an incomplete quadratic equation, which consists in decomposing the left side of it into factors and later using the condition that the product is equal to zero.

For example, 5x ² - 20x \u003d 0. We decompose the left side of the equation into factors, while doing the usual mathematical operation: taking the common factor out of brackets

5x (x - 4) = 0

We use the condition that the products are equal to zero.

5 x = 0 or x - 4 = 0

The answer will be: the first root is 0; the second root is 4.

b) If b \u003d 0, and the free term is not equal to zero, then the equation ax ² + 0x + c \u003d 0 is reduced to an equation of the form ax ² + c \u003d 0. Solve equations in two ways: a) decomposing the polynomial of the equation on the left side into factors ; b) using the properties of the arithmetic square root. Such an equation is solved by one of the methods, for example:

x = ± √ 25/4

x = ± 5/2. The answer is: the first root is 5/2; the second root is - 5/2.

c) If b is equal to 0 and c is equal to 0, then ax² + 0 + 0 = 0 reduces to an equation of the form ax² = 0. In such an equation, x will be equal to 0.

As you can see, incomplete quadratic equations can have at most two roots.