X 7 decision. Solving simple linear equations

In this video, we'll take a look at the whole set. linear equations, which are solved by the same algorithm - that's why they are called the simplest.

To begin with, let's define: what is a linear equation and which of them should be called the simplest?

A linear equation is one in which there is only one variable, and only in the first degree.

The simplest equation means the construction:

All other linear equations are reduced to the simplest ones using the algorithm:

1. Open brackets, if any;
2. Move terms containing a variable to one side of the equal sign, and terms without a variable to the other;
3. Bring like terms to the left and right of the equal sign;
4. Divide the resulting equation by the coefficient of the variable $x$ .

Of course, this algorithm does not always help. The fact is that sometimes, after all these machinations, the coefficient of the variable $x$ turns out to be equal to zero. In this case, two options are possible:

1. The equation has no solutions at all. For example, when you get something like $0\cdot x=8$, i.e. on the left is zero, and on the right is a non-zero number. In the video below, we will look at several reasons why this situation is possible.
2. The solution is all numbers. The only case, when this is possible, the equation is reduced to the construction $0\cdot x=0$. It is quite logical that no matter what $x$ we substitute, it will still turn out “zero is equal to zero”, i.e. correct numerical equality.

And now let's see how it all works on the example of real problems.

Examples of solving equations

Today we deal with linear equations, and only the simplest ones. In general, a linear equation means any equality that contains exactly one variable, and it goes only to the first degree.

Such constructions are solved in approximately the same way:

1. First of all, you need to open the parentheses, if any (as in our last example);
2. Then bring similar
3. Finally, isolate the variable, i.e. everything that is connected with the variable - the terms in which it is contained - is transferred to one side, and everything that remains without it is transferred to the other side.

Then, as a rule, you need to bring similar on each side of the resulting equality, and after that it remains only to divide by the coefficient at "x", and we will get the final answer.

In theory, this looks nice and simple, but in practice, even experienced high school students can make offensive mistakes in fairly simple linear equations. Usually, mistakes are made either when opening brackets, or when counting "pluses" and "minuses".

In addition, it happens that a linear equation has no solutions at all, or so that the solution is the entire number line, i.e. any number. We will analyze these subtleties in today's lesson. But we will start, as you already understood, with the most simple tasks.

Scheme for solving simple linear equations

To begin with, let me once again write the entire scheme for solving the simplest linear equations:

1. Expand the parentheses, if any.
2. Seclude variables, i.e. everything that contains "x" is transferred to one side, and without "x" - to the other.
3. We present similar terms.
4. We divide everything by the coefficient at "x".

Of course, this scheme does not always work, it has certain subtleties and tricks, and now we will get to know them.

Solving real examples of simple linear equations

Task #1

In the first step, we are required to open the brackets. But they are not in this example, so we skip this step. In the second step, we need to isolate the variables. Please note: we are talking only about individual terms. Let's write:

We give similar terms on the left and on the right, but this has already been done here. Therefore, we proceed to the fourth step: divide by a factor:

\[\frac(6x)(6)=-\frac(72)(6)\]

Here we got the answer.

Task #2

In this task, we can observe the brackets, so let's expand them:

Both on the left and on the right, we see approximately the same construction, but let's act according to the algorithm, i.e. sequester variables:

Here are some like:

At what roots does this work? Answer: for any. Therefore, we can write that $x$ is any number.

Task #3

The third linear equation is already more interesting:

\[\left(6-x \right)+\left(12+x \right)-\left(3-2x \right)=15\]

There are several brackets here, but they are not multiplied by anything, they just have different signs in front of them. Let's break them down:

We perform the second step already known to us:

\[-x+x+2x=15-6-12+3\]

Let's calculate:

We perform the last step - we divide everything by the coefficient at "x":

\[\frac(2x)(x)=\frac(0)(2)\]

Things to Remember When Solving Linear Equations

If we ignore too simple tasks, then I would like to say the following:

• As I said above, not every linear equation has a solution - sometimes there are simply no roots;
• Even if there are roots, zero can get in among them - there is nothing wrong with that.

Zero is the same number as the rest, you should not somehow discriminate it or assume that if you get zero, then you did something wrong.

Another feature is related to the expansion of parentheses. Please note: when there is a “minus” in front of them, we remove it, but in brackets we change the signs to opposite. And then we can open it according to standard algorithms: we will get what we saw in the calculations above.

Understanding this simple fact will keep you from making stupid and hurtful mistakes in high school when doing such things is taken for granted.

Solving complex linear equations

Let's move on to more complex equations. Now the constructions will become more complicated and a quadratic function will appear when performing various transformations. However, you should not be afraid of this, because if, according to the author's intention, we solve a linear equation, then in the process of transformation all monomials containing a quadratic function will necessarily be reduced.

Example #1

Obviously, the first step is to open the brackets. Let's do this very carefully:

Now let's take privacy:

\[-x+6((x)^(2))-6((x)^(2))+x=-12\]

Here are some like:

Obviously, this equation has no solutions, so in the answer we write as follows:

\[\variety \]

or no roots.

Example #2

We perform the same steps. First step:

Let's move everything with a variable to the left, and without it - to the right:

Here are some like:

Obviously, this linear equation has no solution, so we write it like this:

\[\varnothing\],

or no roots.

Nuances of the solution

Both equations are completely solved. On the example of these two expressions, we once again made sure that even in the simplest linear equations, everything can be not so simple: there can be either one, or none, or infinitely many. In our case, we considered two equations, in both there are simply no roots.

But I would like to draw your attention to another fact: how to work with brackets and how to expand them if there is a minus sign in front of them. Consider this expression:

Before opening, you need to multiply everything by "x". Please note: multiply each individual term. Inside there are two terms - respectively, two terms and is multiplied.

And only after these seemingly elementary, but very important and dangerous transformations have been completed, can the bracket be opened from the point of view that there is a minus sign after it. Yes, yes: only now, when the transformations are done, we remember that there is a minus sign in front of the brackets, which means that everything down just changes signs. At the same time, the brackets themselves disappear and, most importantly, the front “minus” also disappears.

We do the same with the second equation:

It is no coincidence that I pay attention to these small, seemingly insignificant facts. Because solving equations is always a sequence of elementary transformations, where the inability to clearly and competently perform simple actions leads to the fact that high school students come to me and learn to solve such simple equations again.

Of course, the day will come when you will hone these skills to automatism. You no longer have to perform so many transformations each time, you will write everything in one line. But while you are just learning, you need to write each action separately.

Solving even more complex linear equations

What we are going to solve now can hardly be called the simplest task, but the meaning remains the same.

Task #1

\[\left(7x+1 \right)\left(3x-1 \right)-21((x)^(2))=3\]

Let's multiply all the elements in the first part:

Let's do a retreat:

Here are some like:

Let's do the last step:

\[\frac(-4x)(4)=\frac(4)(-4)\]

Here is our final answer. And, despite the fact that in the process of solving we had coefficients with a quadratic function, however, they mutually annihilated, which makes the equation exactly linear, not square.

Task #2

\[\left(1-4x \right)\left(1-3x \right)=6x\left(2x-1 \right)\]

Let's do the first step carefully: multiply every element in the first bracket by every element in the second. In total, four new terms should be obtained after transformations:

And now carefully perform the multiplication in each term:

Let's move the terms with "x" to the left, and without - to the right:

\[-3x-4x+12((x)^(2))-12((x)^(2))+6x=-1\]

Here are similar terms:

We have received a definitive answer.

Nuances of the solution

The most important remark about these two equations is this: as soon as we start multiplying brackets in which there is more than a term, then this is done according to the following rule: we take the first term from the first and multiply with each element from the second; then we take the second element from the first and similarly multiply with each element from the second. As a result, we get four terms.

On the algebraic sum

With the last example, I would like to remind students what an algebraic sum is. In classical mathematics, by $1-7$ we mean a simple construction: we subtract seven from one. In algebra, we mean by this the following: to the number "one" we add another number, namely "minus seven." This algebraic sum differs from the usual arithmetic sum.

As soon as when performing all the transformations, each addition and multiplication, you begin to see constructions similar to those described above, you simply will not have any problems in algebra when working with polynomials and equations.

In conclusion, let's look at a couple more examples that will be even more complex than the ones we just looked at, and in order to solve them, we will have to slightly expand our standard algorithm.

Solving equations with a fraction

To solve such tasks, one more step will have to be added to our algorithm. But first, I will remind our algorithm:

1. Open brackets.
2. Separate variables.
3. Bring similar.
4. Divide by a factor.

Alas, this wonderful algorithm, for all its efficiency, is not entirely appropriate when we have fractions in front of us. And in what we will see below, we have a fraction on the left and on the right in both equations.

How to work in this case? Yes, it's very simple! To do this, you need to add one more step to the algorithm, which can be performed both before the first action and after it, namely, get rid of fractions. Thus, the algorithm will be as follows:

1. Get rid of fractions.
2. Open brackets.
3. Separate variables.
4. Bring similar.
5. Divide by a factor.

What does it mean to "get rid of fractions"? And why is it possible to do this both after and before the first standard step? In fact, in our case, all fractions are numeric in terms of the denominator, i.e. everywhere the denominator is just a number. Therefore, if we multiply both parts of the equation by this number, then we will get rid of fractions.

Example #1

\[\frac(\left(2x+1 \right)\left(2x-3 \right))(4)=((x)^(2))-1\]

Let's get rid of the fractions in this equation:

\[\frac(\left(2x+1 \right)\left(2x-3 \right)\cdot 4)(4)=\left(((x)^(2))-1 \right)\cdot four\]

Please note: everything is multiplied by “four” once, i.e. just because you have two brackets doesn't mean you have to multiply each of them by "four". Let's write:

\[\left(2x+1 \right)\left(2x-3 \right)=\left(((x)^(2))-1 \right)\cdot 4\]

Now let's open it:

We perform seclusion of a variable:

We carry out the reduction of similar terms:

\[-4x=-1\left| :\left(-4 \right) \right.\]

\[\frac(-4x)(-4)=\frac(-1)(-4)\]

We have received the final solution, we pass to the second equation.

Example #2

\[\frac(\left(1-x \right)\left(1+5x \right))(5)+((x)^(2))=1\]

Here we perform all the same actions:

\[\frac(\left(1-x \right)\left(1+5x \right)\cdot 5)(5)+((x)^(2))\cdot 5=5\]

\[\frac(4x)(4)=\frac(4)(4)\]

Problem solved.

That, in fact, is all that I wanted to tell today.

Key points

The key findings are as follows:

• Know the algorithm for solving linear equations.
• Ability to open brackets.
• Do not worry if somewhere you have quadratic functions, most likely, in the process of further transformations, they will be reduced.
• The roots in linear equations, even the simplest ones, are of three types: one single root, the entire number line is a root, there are no roots at all.

I hope this lesson will help you master a simple, but very important topic for further understanding of all mathematics. If something is not clear, go to the site, solve the examples presented there. Stay tuned, there are many more interesting things waiting for you!

Instruction. For online solutions it is necessary to choose the type of equation and set the dimension of the corresponding matrices.

Example #1. Exercise. Find a solution to a matrix equation
Solution. Denote:
Then the matrix equation will be written in the form: A·X·B = C.
The determinant of matrix A is detA=-1
Since A is a nonsingular matrix, there is an inverse matrix A -1 . Multiply both sides of the equation on the left by A -1: Multiply both sides of this equation on the left by A -1 and on the right by B -1: A -1 A X B B -1 = A -1 C B -1 . Since A A -1 = B B -1 = E and E X = X E = X, then X = A -1 C B -1

inverse matrix A-1:
Find the inverse matrix B -1 .
Transpose matrix B T:
Inverse matrix B -1:
We are looking for the matrix X by the formula: X = A -1 C B -1

Answer:

Example #2. Exercise. Solve matrix equation
Solution. Denote:
Then the matrix equation will be written in the form: A X = B.
The determinant of matrix A is detA=0
Since A is a degenerate matrix (the determinant is 0), therefore, the equation has no solution.

Example #3. Exercise. Find a solution to a matrix equation
Solution. Denote:
Then the matrix equation will be written in the form: X·A = B.
The determinant of matrix A is detA=-60
Since A is a nonsingular matrix, there is an inverse matrix A -1 . Multiply on the right both sides of the equation by A -1: X A A -1 = B A -1 , from which we find that X = B A -1
Find the inverse matrix A -1 .
Transposed matrix A T:
Inverse matrix A -1:
We are looking for the matrix X by the formula: X = B A -1


Answer: >

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This online calculator can

• Correctly carry out standard mathematical functions, written in one line like - 12*3-(7/2) and can handle numbers larger than we count huge numbers in an online calculator. We don’t even know how to call such a number correctly ( there are 34 characters and this is not the limit at all).
• Except tangent, cosine, sinus and other standard functions - the calculator supports calculation operations arc tangent, arc tangent and others.
• Available in the arsenal logarithms, factorials and other cool features
• This online calculator can make charts!!!

To plot graphs, the service uses a special button (gray graph is drawn) or a literal representation of this function (Plot). To build a graph in an online calculator, just write a function: plot(tan(x)),x=-360..360.

We took the simplest plot for the tangent, and after the decimal point, we indicated the range of the X variable from -360 to 360.

You can build absolutely any function, with any number of variables, for example: plot(cos(x)/3z, x=-180..360,z=4) Or even more complex than you can think of. We pay attention to the behavior of the variable X - the interval from and to is indicated using two points.

The only negative (although it is difficult to call it a negative) of this online calculator this is that he does not know how to build spheres and other three-dimensional figures - only a plane.

How to work with the Math Calculator

1. The display (calculator screen) displays the entered expression and the result of its calculation in ordinary characters, as we write on paper. This field is simply for viewing the current operation. The entry is displayed on the display as you type a mathematical expression in the input line.

2. The expression input field is intended for writing the expression to be calculated. It should be noted here that the mathematical symbols used in computer programs, do not always coincide with those that we usually use on paper. In the overview of each function of the calculator, you will find the correct designation for a particular operation and examples of calculations in the calculator. On this page below is a list of all possible operations in the calculator, also indicating their correct spelling.

3. Toolbar are calculator buttons that replace manual input mathematical symbols denoting the corresponding operation. Some calculator buttons (additional functions, unit converter, solution of matrices and equations, graphs) supplement the taskbar with new fields where data for a specific calculation is entered. The "History" field contains examples of writing mathematical expressions, as well as your six most recent entries.

Please note that when you press the buttons for calling additional functions, the converter of values, solving matrices and equations, plotting graphs, the entire calculator panel shifts up, covering part of the display. Fill in the required fields and press the "I" key (highlighted in red in the figure) to see the display in full size.

4. The numeric keypad contains numbers and arithmetic signs. The "C" button deletes the entire entry in the expression input field. To delete characters one by one, you need to use the arrow to the right of the input line.

Try to always close brackets at the end of an expression. For most operations, this is not critical, the online calculator will calculate everything correctly. However, in some cases errors are possible. For example, when raising to a fractional power, unclosed brackets will cause the denominator of the fraction in the exponent to go to the denominator of the base. On the display, the closing bracket is indicated in pale gray, it must be closed when the recording is completed.

I. ax 2 \u003d 0 – incomplete quadratic equation (b=0, c=0 ). Solution: x=0. Answer: 0.

Solve equations.

2x·(x+3)=6x-x 2 .

Solution. Expand the brackets by multiplying 2x for each term in brackets:

2x2 +6x=6x-x2 ; moving the terms from the right side to the left side:

2x2 +6x-6x+x2=0; Here are similar terms:

3x 2 =0, hence x=0.

Answer: 0.

II. ax2+bx=0 –incomplete quadratic equation (s=0 ). Solution: x (ax+b)=0 → x 1 =0 or ax+b=0 → x 2 =-b/a. Answer: 0; -b/a.

5x2 -26x=0.

Solution. Take out the common factor X for brackets:

x(5x-26)=0; each factor can be zero:

x=0 or 5x-26=0→ 5x=26, divide both sides of the equality by 5 and we get: x \u003d 5.2.

Answer: 0; 5,2.

Example 3 64x+4x2=0.

Solution. Take out the common factor 4x for brackets:

4x(16+x)=0. We have three factors, 4≠0, therefore, or x=0 or 16+x=0. From the last equality we get x=-16.

Answer: -16; 0.

Example 4(x-3) 2 +5x=9.

Solution. Applying the formula for the square of the difference of two expressions, open the brackets:

x 2 -6x+9+5x=9; transform to the form: x 2 -6x+9+5x-9=0; Here are similar terms:

x2-x=0; endure X outside the brackets, we get: x (x-1)=0. From here or x=0 or x-1=0→ x=1.

Answer: 0; 1.

III. ax2+c=0 –incomplete quadratic equation (b=0 ); Solution: ax 2 \u003d -c → x 2 \u003d -c / a.

If a (-c/a)<0 , then there are no real roots. If a (-s/a)>0

Example 5 x 2 -49=0.

Solution.

x 2 \u003d 49, from here x=±7. Answer:-7; 7.

Example 6 9x2-4=0.

Solution.

Often you need to find the sum of squares (x 1 2 + x 2 2) or the sum of cubes (x 1 3 + x 2 3) of the roots of a quadratic equation, less often - the sum of the reciprocals of the squares of the roots or the sum of arithmetic square roots from the roots of the quadratic equation:

Vieta's theorem can help with this:

x 2 +px+q=0

x 1 + x 2 \u003d-p; x 1 ∙ x 2 \u003d q.

Express through p and q:

1) the sum of the squares of the roots of the equation x2+px+q=0;

2) the sum of the cubes of the roots of the equation x2+px+q=0.

Solution.

1) Expression x 1 2 + x 2 2 obtained by squaring both sides of the equation x 1 + x 2 \u003d-p;

(x 1 +x 2) 2 \u003d (-p) 2; open the brackets: x 1 2 +2x 1 x 2 + x 2 2 =p 2; we express the desired amount: x 1 2 +x 2 2 \u003d p 2 -2x 1 x 2 \u003d p 2 -2q. We have a useful equation: x 1 2 +x 2 2 \u003d p 2 -2q.

2) Expression x 1 3 + x 2 3 represent by the formula of the sum of cubes in the form:

(x 1 3 +x 2 3)=(x 1 +x 2)(x 1 2 -x 1 x 2 +x 2 2)=-p (p 2 -2q-q)=-p (p 2 -3q).

Another useful equation: x 1 3 + x 2 3 \u003d-p (p 2 -3q).

Examples.

3) x 2 -3x-4=0. Without solving the equation, calculate the value of the expression x 1 2 + x 2 2.

Solution.

x 1 + x 2 \u003d-p \u003d 3, and the work x 1 ∙x 2 \u003d q \u003din example 1) equality:

x 1 2 +x 2 2 \u003d p 2 -2q. We have -p=x 1 +x 2 = 3 → p 2 =3 2 =9; q= x 1 x 2 = -4. Then x 1 2 + x 2 2 =9-2 (-4)=9+8=17.

Answer: x 1 2 + x 2 2 =17.

4) x 2 -2x-4=0. Calculate: x 1 3 +x 2 3 .

Solution.

By Vieta's theorem, the sum of the roots of this reduced quadratic equation x 1 + x 2 \u003d-p \u003d 2, and the work x 1 ∙x 2 \u003d q \u003d-four. Let us apply what we have obtained ( in example 2) equality: x 1 3 +x 2 3 \u003d-p (p 2 -3q) \u003d 2 (2 2 -3 (-4))=2 (4+12)=2 16=32.

Answer: x 1 3 + x 2 3 =32.

Question: what if we are given a non-reduced quadratic equation? Answer: it can always be “reduced” by dividing term by term by the first coefficient.

5) 2x2 -5x-7=0. Without solving, calculate: x 1 2 + x 2 2.

Solution. We are given a complete quadratic equation. Divide both sides of the equation by 2 (the first coefficient) and get the following quadratic equation: x 2 -2.5x-3.5 \u003d 0.

By Vieta's theorem, the sum of the roots is 2,5 ; the product of the roots is -3,5 .

We solve in the same way as an example 3) using the equality: x 1 2 +x 2 2 \u003d p 2 -2q.

x 1 2 +x 2 2 =p 2 -2q= 2,5 2 -2∙(-3,5)=6,25+7=13,25.

Answer: x 1 2 + x 2 2 = 13,25.

6) x2 -5x-2=0. Find:

Let us transform this equality and, by replacing the sum of the roots in terms of the Vieta theorem, -p, and the product of the roots through q, we get another useful formula. When deriving the formula, we used equality 1): x 1 2 +x 2 2 \u003d p 2 -2q.

In our example x 1 + x 2 \u003d -p \u003d 5; x 1 ∙x 2 \u003d q \u003d-2. Substitute these values ​​into the resulting formula:

7) x 2 -13x+36=0. Find:

Let's transform this sum and get a formula by which it will be possible to find the sum of arithmetic square roots from the roots of a quadratic equation.

We have x 1 + x 2 \u003d -p \u003d 13; x 1 ∙x 2 \u003d q \u003d 36. Substitute these values ​​into the derived formula:

Advice : always check the possibility of finding the roots of a quadratic equation in a suitable way, because 4 reviewed useful formulas allow you to quickly complete the task, first of all, in cases where the discriminant is an “inconvenient” number. In all simple cases, find the roots and operate on them. For example, in the last example, we select the roots using the Vieta theorem: the sum of the roots should be equal to 13 , and the product of the roots 36 . What are these numbers? Of course, 4 and 9. Now calculate the sum of the square roots of these numbers: 2+3=5. That's it!

I. Vieta's theorem for the reduced quadratic equation.

The sum of the roots of the reduced quadratic equation x 2 +px+q=0 is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term:

x 1 + x 2 \u003d-p; x 1 ∙ x 2 \u003d q.

Find the roots of the given quadratic equation using Vieta's theorem.

Example 1) x 2 -x-30=0. This is the reduced quadratic equation ( x 2 +px+q=0), the second coefficient p=-1, and the free term q=-30. First, make sure that the given equation has roots, and that the roots (if any) will be expressed as integers. For this, it is sufficient that the discriminant be the full square of an integer.

Finding the discriminant D=b 2 - 4ac=(-1) 2 -4∙1∙(-30)=1+120=121= 11 2 .

Now, according to the Vieta theorem, the sum of the roots must be equal to the second coefficient, taken with the opposite sign, i.e. ( -p), and the product is equal to the free term, i.e. ( q). Then:

x 1 + x 2 =1; x 1 ∙ x 2 \u003d -30. We need to choose such two numbers so that their product is equal to -30 , and the sum is unit. These are the numbers -5 and 6 . Answer: -5; 6.

Example 2) x 2 +6x+8=0. We have the reduced quadratic equation with the second coefficient p=6 and free member q=8. Make sure that there are integer roots. Let's find the discriminant D1 D1=3 2 -1∙8=9-8=1=1 2 . The discriminant D 1 is the perfect square of the number 1 , so the roots of this equation are integers. We choose the roots according to the Vieta theorem: the sum of the roots is equal to –p=-6, and the product of the roots is q=8. These are the numbers -4 and -2 .

Actually: -4-2=-6=-p; -4∙(-2)=8=q. Answer: -4; -2.

Example 3) x 2 +2x-4=0. In this reduced quadratic equation, the second coefficient p=2, and the free term q=-4. Let's find the discriminant D1, since the second coefficient is an even number. D1=1 2 -1∙(-4)=1+4=5. The discriminant is not a perfect square of a number, so we do conclusion: the roots of this equation are not integers and cannot be found using Vieta's theorem. So, we solve this equation, as usual, according to the formulas (in this case, according to the formulas). We get:

Example 4). Write a quadratic equation using its roots if x 1 \u003d -7, x 2 \u003d 4.

Solution. The desired equation will be written in the form: x 2 +px+q=0, moreover, based on the Vieta theorem –p=x1 +x2=-7+4=-3 →p=3; q=x 1 ∙x 2=-7∙4=-28 . Then the equation will take the form: x2 +3x-28=0.

Example 5). Write a quadratic equation using its roots if:

II. Vieta's theorem for the complete quadratic equation ax2+bx+c=0.

The sum of the roots is minus b divided by a, the product of the roots is With divided by a:

x 1 + x 2 \u003d -b / a; x 1 ∙ x 2 \u003d c / a.

Example 6). Find the sum of the roots of a quadratic equation 2x2 -7x-11=0.

Solution.

We are convinced that this equation will have roots. To do this, it is enough to write an expression for the discriminant, and without calculating it, just make sure that the discriminant is greater than zero. D=7 2 -4∙2∙(-11)>0 . And now let's use theorem Vieta for complete quadratic equations.

x 1 + x 2 =-b:a=- (-7):2=3,5.

Example 7). Find the product of the roots of a quadratic equation 3x2 +8x-21=0.

Solution.

Let's find the discriminant D1, since the second coefficient ( 8 ) is an even number. D1=4 2 -3∙(-21)=16+63=79>0 . The quadratic equation has 2 root, according to the Vieta theorem, the product of the roots x 1 ∙ x 2 \u003d c: a=-21:3=-7.

I. ax 2 +bx+c=0 is a general quadratic equation

Discriminant D=b 2 - 4ac.

If a D>0, then we have two real roots:

If a D=0, then we have a single root (or two equal roots) x=-b/(2a).

If D<0, то действительных корней нет.

Example 1) 2x2 +5x-3=0.

Solution. a=2; b=5; c=-3.

D=b 2-4ac=5 2 -4∙2∙(-3)=25+24=49=7 2 >0; 2 real roots.

4x2 +21x+5=0.

Solution. a=4; b=21; c=5.

D=b 2-4ac=21 2 - 4∙4∙5=441-80=361=19 2 >0; 2 real roots.

II. ax2+bx+c=0 – special quadratic equation for an even second

coefficient b

Example 3) 3x2 -10x+3=0.

Solution. a=3; b\u003d -10 (even number); c=3.

Example 4) 5x2-14x-3=0.

Solution. a=5; b= -14 (even number); c=-3.

Example 5) 71x2 +144x+4=0.

Solution. a=71; b=144 (even number); c=4.

Example 6) 9x 2 -30x+25=0.

Solution. a=9; b\u003d -30 (even number); c=25.

III. ax2+bx+c=0 – quadratic equation private type, provided: a-b+c=0.

The first root is always minus one, and the second root is minus With divided by a:

x 1 \u003d -1, x 2 \u003d - c / a.

Example 7) 2x2+9x+7=0.

Solution. a=2; b=9; c=7. Let's check the equality: a-b+c=0. We get: 2-9+7=0 .

Then x 1 \u003d -1, x 2 \u003d -c / a \u003d -7 / 2 \u003d -3.5. Answer: -1; -3,5.

IV. ax2+bx+c=0 – quadratic equation of a particular form under the condition : a+b+c=0.

The first root is always equal to one, and the second root is equal to With divided by a:

x 1 \u003d 1, x 2 \u003d c / a.

Example 8) 2x2 -9x+7=0.

Solution. a=2; b=-9; c=7. Let's check the equality: a+b+c=0. We get: 2-9+7=0 .

Then x 1 \u003d 1, x 2 \u003d c / a \u003d 7/2 \u003d 3.5. Answer: 1; 3,5.

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1. Solution of the system by the substitution method.
2. Solution of the system by term-by-term addition (subtraction) of the equations of the system.

In order to solve the system of equations substitution method you need to follow a simple algorithm:
1. We express. From any equation, we express one variable.
2. Substitute. We substitute in another equation instead of the expressed variable, the resulting value.
3. We solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) need:
1. Select a variable for which we will make the same coefficients.
2. We add or subtract the equations, as a result we get an equation with one variable.
3. We solve the resulting linear equation. We find a solution to the system.

The solution of the system is the intersection points of the graphs of the function.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, hence it turns out that it is easiest to express the variable x from the second equation.
x=3+10y

2. After expressing, we substitute 3 + 10y in the first equation instead of the variable x.
2(3+10y)+5y=1

3. We solve the resulting equation with one variable.
2(3+10y)+5y=1 (open brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution of the system of equations is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first paragraph where we expressed we substitute y there.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place, we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve by term-by-term addition (subtraction).

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. Select a variable, let's say we select x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. From the first equation, subtract the second to get rid of the variable x. We solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y in any of the equations, let's say in the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The point of intersection will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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