Quadratic equation - easy to solve! *Hereinafter referred to as “KU”. Friends, it would seem that there could be nothing simpler in mathematics than solving such an equation. But something told me that many people have problems with him. I decided to see how many on-demand impressions Yandex gives out per month. Here's what happened, look:

What does it mean? This means that about 70,000 people a month are searching for this information, what does this summer have to do with it, and what will happen among school year— there will be twice as many requests. This is not surprising, because those guys and girls who graduated from school a long time ago and are preparing for the Unified State Exam are looking for this information, and schoolchildren also strive to refresh their memory.

Despite the fact that there are a lot of sites that tell you how to solve this equation, I decided to also contribute and publish the material. Firstly, I would like visitors to come to my site based on this request; secondly, in other articles, when the topic of “KU” comes up, I will provide a link to this article; thirdly, I’ll 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 c are arbitrary numbers, with a≠0.

In the school course, the material is given in the following form - the equations are divided into three classes:

1. They have two roots.

2. *Have only one root.

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

How are roots calculated? Just!

We calculate the discriminant. Underneath this “terrible” word lies a very simple formula:

The root formulas are as follows:

*You need to know these formulas 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:

In this regard, when the discriminant is equal to zero, the school course says that one root is obtained, here it is equal to nine. Everything is correct, it is so, but...

This idea is somewhat incorrect. In fact, there are two roots. Yes, yes, don’t be surprised, you get two equal roots, and to be mathematically precise, then the answer should write two roots:

x 1 = 3 x 2 = 3

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

Now the next example:

As we know, the root of a negative number cannot be taken, so the solutions in in this case No.

That's the whole decision process.

Quadratic function.

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

This is a function of the form:

where x and y are variables

a, b, c – given numbers, with 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) and none (the discriminant is negative). Details about quadratic function You can view article by Inna Feldman.

Let's look at examples:

Example 1: Solve 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

*It was possible to immediately divide the left and right sides of the equation by 2, that is, simplify it. Calculations will be easier.

Example 2: Decide x 2–22 x+121 = 0

a=1 b=–22 c=121

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

We found that x 1 = 11 and x 2 = 11

It is permissible to write x = 11 in the answer.

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 little 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 – this is a SINGLE NUMBER, not an addition.

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

Now consider the equation:

We get two conjugate roots.

Incomplete quadratic equation.

Let's consider special cases, this is when the coefficient “b” or “c” is equal to zero (or both are equal to zero). They can be solved easily without any discriminatory issues.

Case 1. Coefficient b = 0.

The equation becomes:

Let's convert:

Example:

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

Case 2. Coefficient c = 0.

The equation becomes:

Let's transform and 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 you to solve equations with large coefficients.

Ax 2 + bx+ c=0 equality holds

a + b+ c = 0, That

- if for the coefficients of the equation Ax 2 + bx+ c=0 equality holds

a+ s =b, That

These properties help solve a certain type of equation.

Example 1: 5001 x 2 –4995 x – 6=0

The sum of the odds is 5001+( – 4995)+(– 6) = 0, which means

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

Equality holds a+ s =b, Means

Regularities of coefficients.

1. If in the equation ax 2 + bx + c = 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 equal

ax 2 + (a 2 +1)∙x+ a= 0 = > x 1 = –a x 2 = –1/a.

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

x 1 = –6 x 2 = –1/6.

2. If in the equation ax 2 – bx + c = 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 equal

ax 2 – (a 2 +1)∙x+ a= 0 = > x 1 = a x 2 = 1/a.

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

x 1 = 15 x 2 = 1/15.

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

ax 2 + (a 2 –1)∙x – a= 0 = > x 1 = – a x 2 = 1/a.

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

x 1 = – 17 x 2 = 1/17.

4. If in the equation ax 2 – bx – c = 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 equal

ax 2 – (a 2 –1)∙x – a= 0 = > x 1 = a x 2 = – 1/a.

Example. Consider the equation 10x 2 – 99x –10 = 0.

x 1 = 10 x 2 = – 1/10

Vieta's theorem.

Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, we 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 total, the number 14 gives only 5 and 9. These are roots. With a certain skill, using the presented theorem, you can solve many quadratic equations orally immediately.

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

TRANSPORTATION METHOD

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

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

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

Using Vieta's theorem in equation (2), it is easy to determine that x 1 = 10 x 2 = 1

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

x 1 = 5 x 2 = 0.5.

What is the rationale? Look what's happening.

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

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

The second (modified) one has roots that are 2 times larger.

Therefore, we divide the result by 2.

*If we roll three, we will divide the result by 3, etc.

Answer: x 1 = 5 x 2 = 0.5

Sq. ur-ie and Unified State Examination.

I’ll tell you briefly about its importance - YOU MUST BE ABLE TO DECIDE quickly and without thinking, you need to know the formulas of roots and discriminants by heart. Many of the problems included in the Unified State Examination tasks come down to solving a quadratic equation (geometric ones included).

Something worth noting!

1. The form of writing an 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 quantity and it can be denoted by any other letter - t, q, p, h and others.

Quadratic equations are studied in 8th grade, so there is nothing complicated here. The ability to solve them is absolutely necessary.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, note that all quadratic equations can be divided into three classes:

1. They have no roots;
2. Have exactly one root;
3. They have two different roots.

This is an important difference between quadratic equations and linear ones, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac.

You need to know this formula by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people believe. Take a look at the examples and you will understand everything yourself:

Task. How many roots do quadratic equations have:

1. x 2 − 8x + 12 = 0;
2. 5x 2 + 3x + 7 = 0;
3. x 2 − 6x + 9 = 0.

Let's write out the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 − 4 5 7 = 9 − 140 = −131.

The discriminant is negative, there are no roots. The last equation left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is zero - the root will be one.

Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.

Roots of a quadratic equation

Now let's move on to the solution itself. If the discriminant D > 0, the roots can be found using the formulas:

Basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you will get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

1. x 2 − 2x − 3 = 0;
2. 15 − 2x − x 2 = 0;
3. x 2 + 12x + 36 = 0.

First equation:
x 2 − 2x − 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 · (−1) · 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of errors.

Incomplete quadratic equations

It happens that a quadratic equation is slightly different from what is given in the definition. For example:

1. x 2 + 9x = 0;
2. x 2 − 16 = 0.

It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating the discriminant. So, let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.

Let's consider the remaining cases. Let b = 0, then we obtain an incomplete quadratic equation of the form ax 2 + c = 0. Let us transform it a little:

Since arithmetic Square root exists only from a non-negative number, the last equality makes sense only for (−c /a) ≥ 0. Conclusion:

1. If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c /a) ≥ 0 is satisfied, there will be two roots. The formula is given above;
2. If (−c /a)< 0, корней нет.

As you can see, a discriminant was not required—there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c /a) ≥ 0. It is enough to express the value x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If it is negative, there will be no roots at all.

Now let's look at equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor the polynomial:

The product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these equations:

Task. Solve quadratic equations:

1. x 2 − 7x = 0;
2. 5x 2 + 30 = 0;
3. 4x 2 − 9 = 0.

x 2 − 7x = 0 ⇒ x · (x − 7) = 0 ⇒ x 1 = 0; x 2 = −(−7)/1 = 7.

5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, because a square cannot be equal to a negative number.

4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.

Quadratic equations are studied in 8th grade, so there is nothing complicated here. The ability to solve them is absolutely necessary.

is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, note that all quadratic equations can be divided into three classes:

1. They have no roots;
2. Have exactly one root;
3. They have two different roots.

This is an important difference between quadratic equations and linear ones, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then it is simply the number D = b 2 − 4ac.

You need to know this formula by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people believe. Take a look at the examples and you will understand everything yourself:

Task. How many roots do quadratic equations have:

1. x 2 − 8x + 12 = 0;
2. 5x 2 + 3x + 7 = 0;
3. x 2 − 6x + 9 = 0.

Let's write out the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 − 4 5 7 = 9 − 140 = −131.

The discriminant is negative, there are no roots. The last equation left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is zero - the root will be one.

Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.

Roots of a quadratic equation

Now let's move on to the solution itself. If the discriminant D > 0, the roots can be found using the formulas:

When D = 0, you can use any of these formulas - you will get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

Solve quadratic equations:

1. x 2 − 2x − 3 = 0;
2. 15 − 2x − x 2 = 0;
3. x 2 + 12x + 36 = 0.

First equation:
x 2 − 2x − 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 · (−1) · 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of errors.

Incomplete quadratic equations

It happens that a quadratic equation is slightly different from what is given in the definition. For example:

1. x 2 + 9x = 0;
2. x 2 − 16 = 0.

It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating the discriminant. So, let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.

Let's consider the remaining cases. Let b = 0, then we obtain an incomplete quadratic equation of the form ax 2 + c = 0. Let us transform it a little:

Since the arithmetic square root exists only of a non-negative number, the last equality only makes sense when (−c/a) ≥ 0. Conclusion:

1. If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c/a) ≥ 0 is satisfied, there will be two roots. The formula is given above;
2. If (−c/a)< 0, корней нет.

As you can see, a discriminant was not required—there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c/a) ≥ 0.

Solving incomplete quadratic equations.

It is enough to express the value x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If it is negative, there will be no roots at all.

Now let's look at equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor the polynomial:

The product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these equations:

Task. Solve quadratic equations:

1. x 2 − 7x = 0;
2. 5x 2 + 30 = 0;
3. 4x 2 − 9 = 0.

x 2 − 7x = 0 ⇒ x · (x − 7) = 0 ⇒ x 1 = 0; x 2 = −(−7)/1 = 7.

5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, because a square cannot be equal to a negative number.

4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.

See also:

The quadratic equation is ax 2 + bx + c = 0.

Incomplete quadratic equations are equations of three types:

• ax 2 + bx = 0 when coefficient c = 0.
• ax 2 + c = 0 when coefficient b = 0.
• ax 2 = 0 when both b and c are 0.

Coefficient a, by definition of a quadratic equation, cannot be equal to zero.

Incomplete quadratic equations are easier to solve than complete quadratic equations. Solution methods vary depending on the type of incomplete quadratic equation.

The easiest way to solve equations is ax 2 = 0. If a, by definition of a quadratic equation, cannot be equal to zero, then it is obvious that only x 2, and therefore x itself, can be equal to zero. Equations of this type always have one root, it is equal to 0.

Incomplete quadratic equations. Solving incomplete quadratic equations

For example:

–3x 2 = 0
x 2 = 0/–3
x 2 = 0
x = √0
x = 0

Equations of the form ax 2 + c = 0 are transformed to the form ax 2 = –c and solved similarly to the previous one. However, there are either two roots or more than one.

ax 2 + c = 0
ax 2 = –c
x 2 = –c/a
x = √(–c/a)

Here, if the radical expression is negative, then the equation has no roots. If positive, then there will be two roots: √(–c/a) and –√(–c/a). An example of solving such an equation:

4x 2 – 16 = 0
4x2 = 16
x 2 = 16 / 4
x 2 = 4
x = √4
x 1 = 2; x 2 = –2

Incomplete quadratic equations of the form ax 2 + bx = 0 are solved by taking the common factor out of brackets. In this case it is x. The resulting equation is x(ax + b) = 0. This equation has two roots: either x = 0 or ax + b = 0. Solving the second equation we get x = –b/a. Thus, equations of the form ax 2 + bx = 0 have two roots: x 1 = 0, x 2 = –b/a. An example of solving such an equation:

3x 2 – 10x = 0
x(3x – 10) = 0
x 1 = 0; x 2 = 10/3 = 3,(33)

Finding the roots of a quadratic equation 8th grade

Formula
The roots of the quadratic equation ax 2 + bx + c = 0 can be found by
formula: , Where — discriminant

quadratic equation.

There are three possible rules:

Rule 1
1. D > 0.

8.2.1. Solving incomplete quadratic equations

Then the equation has 2 different roots:

Example
2x 2 + 7x – 4 = 0;

a = 2, b = 7, c = -4.

D = 7 2 - 4 2 (- 4) = 81 > 0,

x 1 = -7 — ? 81 2 2 = — 4;

x 2 = -7 + ? 81 2 2 = 1 2 .

Rule 2
2. D = 0. Then the equation has the only one root.

Example
x 2 – 4x + 4 = 0.

D = (-4) 2 — 4 1 4 = 0, x = — -4 2 1 = 2.

Note that x 2 – 4x + 4 = 0 x = 2.

Rule 3
3. D

Example
3x 2 - x + 7 = 0.

D = (-1) 2 - 4 3 7 = -83

With an even second coefficient

Rule, formulas
If b = 2k, then the roots of the equation ax + 2kx + c = 0 are found by the formula:

Example 1
1. x + 18x + 32 = 0.

a = 1; b = 18 => k = b2 = 9; c = 32.

D 1 = D4 = (18 2 ) 2 - 1 32 = 49 > 0, which means the equation has 2 roots:

x 1 = -9 -? 49 1 = -16, x 2 = -9 + 7 = -2.

Example 2
2. 3x 2 + 2x + 1 = 0.

a = 3; b2 = 1; c = 1.

D 1 = D4 = 1 2 — 1 3 = -2

Example 3
3. 196x 2 + 28x + 1 = 0.

a = 196; b2 = -14; c = 1.

D 1 = D4 = (- 14) 2 - 196 = 0, which means the equation has one root.

x = 14 196 = 1 14 .

Abbreviated multiplication formulas

Abbreviated multiplication formulas.

— Study of abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; cube of the sum and cube of the difference of two expressions; sums and differences of cubes of two expressions.

— Application of abbreviated multiplication formulas when solving examples.

To simplify expressions, factor polynomials, and reduce polynomials to standard form, abbreviated multiplication formulas are used.

Solving Quadratic Equations

Abbreviated multiplication formulas need to be known by heart.

Let a, b R. Then:

1. The square of the sum of two expressions is equal to the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression.

(a + b) 2 = a 2 + 2ab + b 2

2. The square of the difference of two expressions is equal to the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression.

(a - b) 2 = a 2 - 2ab + b 2

3. Difference of squares two expressions is equal to the product of the difference of these expressions and their sum.

a 2 - b 2 = (a -b) (a+b)

4. Cube of sum two expressions is equal to the cube of the first expression plus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second plus the cube of the second expression.

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

5. Difference cube two expressions is equal to the cube of the first expression minus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second minus the cube of the second expression.

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

6. Sum of cubes two expressions is equal to the product of the sum of the first and second expressions and the incomplete square of the difference of these expressions.

a 3 + b 3 = (a + b) (a 2 - ab + b 2)

7. Difference of cubes two expressions is equal to the product of the difference of the first and second expressions by the incomplete square of the sum of these expressions.

Application of abbreviated multiplication formulas when solving examples.

Example 1.

Calculate

a) Using the formula for the square of the sum of two expressions, we have

(40+1) 2 = 40 2 + 2 40 1 + 1 2 = 1600 + 80 + 1 = 1681

b) Using the formula for the square of the difference of two expressions, we obtain

98 2 = (100 – 2) 2 = 100 2 – 2 100 2 + 2 2 = 10000 – 400 + 4 = 9604

Example 2.

Calculate

Using the formula for the difference of the squares of two expressions, we get

Example 3.

Simplify an expression

(x - y) 2 + (x + y) 2

Let's use the formulas for the square of the sum and the square of the difference of two expressions

(x - y) 2 + (x + y) 2 = x 2 - 2xy + y 2 + x 2 + 2xy + y 2 = 2x 2 + 2y 2

Abbreviated multiplication formulas in one table:

(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
a 2 - b 2 = (a - b) (a+b)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
a 3 - b 3 = (a - b) (a 2 + ab + b 2)

1. Selecting a complete square. Formulas for the roots of a quadratic equation.
2. Examples of solving quadratic equations.
3. Solving incomplete quadratic equations.
4. Factoring a quadratic trinomial.

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

There is nothing complicated in determining the type of an incomplete polynomial. It is best to consider the main differences using visual examples:

1. If b = 0, then the equation is ax 2 + c = 0.
2. If c = 0, then the expression ax 2 + bx = 0 should be solved.
3. If b = 0 and c = 0, then the polynomial turns into an equality like ax 2 = 0.

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

General algorithm for searching variables and examples with solutions

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

1. Reduce the expression to a form convenient for finding roots.
2. Perform calculations.
3. Write down the answer.

The easiest way to solve incomplete equations is to factor the left side and leave a zero on the right. Thus, the formula for an incomplete quadratic equation for finding 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 specific example finding the roots of an incomplete equation:

As you can see, in this case b = 0. Let’s 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. The values ​​of the variable x1 = 0.5 and (or) x2 = -0.5 meet similar requirements.

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

If there is no free term in the expression, the problem is greatly simplified. It will be enough just to find and bracket 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.

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

Traditional solution method and incomplete quadratic equations

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

7x 2 – 3x = 0.

Let's calculate the discriminant value: D = (-3)2 – 4 ⋅ (-7) ⋅ 0 = 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) = 0,

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

As you can see, both methods give the same result, but solving the equation using the second method was much easier and faster.

Vieta's theorem

But what to do with Vieta’s favorite theorem? Can this method be used when the trinomial is incomplete? Let's try to understand the aspects of bringing 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 general appearance, 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 the development of skills when solving specific problems. Let's turn again to the reference book of standard tasks for the Unified State Exam and find a suitable example:

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

x 2 + 0 – 16 = 0.

The next step is to create a system of conditions:

Obviously, the roots of the quadratic polynomial will be x 1 = 4 and x 2 = -4.

Now, let's practice bringing the equation to its general form. Let's take the following example: 1/4× x 2 – 1 = 0

In order to apply Vieta's theorem to an expression, it is necessary to get rid of the fraction. Let’s 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 Vieta’s theorem, but it is much easier and faster to get the answer by simply moving c = 4 to the right side of the equation: x2 = 4.

To summarize, it should be said that the best way Solving incomplete equations by factoring is the simplest and fastest method. If difficulties arise in the process of searching for roots, you can turn to the traditional method of finding roots through a discriminant.

In this article we will look at solving incomplete quadratic equations.

But first, let’s repeat what equations are called quadratic. An equation of the form ax 2 + bx + c = 0, where x is a variable, and the coefficients a, b and c are some numbers, and a ≠ 0, is called square. As we see, the coefficient for x 2 is not equal to zero, and therefore the coefficients for x or the free term can be equal to zero, in which case we get an incomplete quadratic equation.

There are three types of incomplete quadratic equations:

1) If b = 0, c ≠ 0, then ax 2 + c = 0;

2) If b ≠ 0, c = 0, then ax 2 + bx = 0;

3) If b = 0, c = 0, then ax 2 = 0.

• Let's figure out how to solve equations of the form ax 2 + c = 0.

To solve the equation, we move the free term c to the right side of the equation, we get

ax 2 = ‒s. Since a ≠ 0, we divide both sides of the equation by a, then x 2 = ‒c/a.

If ‒с/а > 0, then the equation has two roots

x = ±√(–c/a) .

If ‒c/a< 0, то это уравнение решений не имеет. Более наглядно решение данных уравнений представлено на схеме.

Let's try to understand with examples how to solve such equations.

Example 1. Solve the equation 2x 2 ‒ 32 = 0.

Answer: x 1 = - 4, x 2 = 4.

Example 2. Solve the equation 2x 2 + 8 = 0.

Answer: the equation has no solutions.

• Let's figure out how to solve it equations of the form ax 2 + bx = 0.

To solve the equation ax 2 + bx = 0, let's factorize it, that is, take x out of brackets, we get x(ax + b) = 0. The product is equal to zero if at least one of the factors is equal to zero. Then either x = 0, or ax + b = 0. Solving the equation ax + b = 0, we get ax = - b, whence x = - b/a. An equation of the form ax 2 + bx = 0 always has two roots x 1 = 0 and x 2 = ‒ b/a. See what the solution to equations of this type looks like in the diagram.

Let's consolidate our knowledge with a specific example.

Example 3. Solve the equation 3x 2 ‒ 12x = 0.

x(3x ‒ 12) = 0

x= 0 or 3x – 12 = 0

Answer: x 1 = 0, x 2 = 4.

• Equations of the third type ax 2 = 0 are solved very simply.

If ax 2 = 0, then x 2 = 0. The equation has two equal roots x 1 = 0, x 2 = 0.

For clarity, let's look at the diagram.

Let us make sure when solving Example 4 that equations of this type can be solved very simply.

Example 4. Solve the equation 7x 2 = 0.

Answer: x 1, 2 = 0.

It is not always immediately clear what type of incomplete quadratic equation we have to solve. Consider the following example.

Example 5. Solve the equation

Let's multiply both sides of the equation by a common denominator, that is, by 30

Let's cut it down

5(5x 2 + 9) – 6(4x 2 – 9) = 90.

Let's open the brackets

25x 2 + 45 – 24x 2 + 54 = 90.

Let's give similar

Let's move 99 from the left side of the equation to the right, changing the sign to the opposite

Answer: no roots.

We looked at how incomplete quadratic equations are solved. I hope that now you will not have any difficulties with such tasks. Be careful when determining the type of incomplete quadratic equation, then you will succeed.

If you have questions on this topic, sign up for my lessons, we will solve the problems that arise together.

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