Quadratic equations. Discriminant. Solution, examples.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Types of quadratic equations
What's happened quadratic equation? What does it look like? In term quadratic equation the keyword is "square". This means that in the equation Necessarily there must be an x squared. In addition to it, the equation may (or may not!) contain just X (to the first power) and just a number (free member). And there should be no X's to a power greater than two.
Speaking mathematical language, a quadratic equation is an equation of the form:
Here a, b and c- some numbers. b and c- absolutely any, but A– anything other than zero. For example:
Here A =1; b = 3; c = -4
Here A =2; b = -0,5; c = 2,2
Here A =-3; b = 6; c = -18
Well, you understand...
In these quadratic equations on the left there is full set members. X squared with a coefficient A, x to the first power with coefficient b And free member s.
Such quadratic equations are called full.
And if b= 0, what do we get? We have X will be lost to the first power. This happens when multiplied by zero.) It turns out, for example:
5x 2 -25 = 0,
2x 2 -6x=0,
-x 2 +4x=0
And so on. And if both coefficients b And c are equal to zero, then it’s even simpler:
2x 2 =0,
-0.3x 2 =0
Such equations where something is missing are called incomplete quadratic equations. Which is quite logical.) Please note that x squared is present in all equations.
By the way, why A can't be equal to zero? And you substitute instead A zero.) Our X squared will disappear! The equation will become linear. And the solution is completely different...
That's all the main types of quadratic equations. Complete and incomplete.
Solving quadratic equations.
Solving complete quadratic equations.
Quadratic equations are easy to solve. According to formulas and clear, simple rules. At the first stage it is necessary given equation lead to a standard form, i.e. to the form:
If the equation is already given to you in this form, you do not need to do the first stage.) The main thing is to correctly determine all the coefficients, A, b And c.
The formula for finding the roots of a quadratic equation looks like this:
The expression under the root sign is called discriminant. But more about him below. As you can see, to find X, we use only a, b and c. Those. coefficients from a quadratic equation. Just carefully substitute the values a, b and c We calculate into this formula. Let's substitute with your own signs! For example, in the equation:
A =1; b = 3; c= -4. Here we write it down:
The example is almost solved:
This is the answer.
Everything is very simple. And what, you think it’s impossible to make a mistake? Well, yes, how...
The most common mistakes are confusion with sign values a, b and c. Or rather, not with their signs (where to get confused?), but with the substitution of negative values into the formula for calculating the roots. What helps here is a detailed recording of the formula with specific numbers. If there are problems with calculations, do that!
Suppose we need to solve the following example:
Here a = -6; b = -5; c = -1
Let's say you know that you rarely get answers the first time.
Well, don't be lazy. It will take about 30 seconds to write an extra line. And the number of errors will decrease sharply. So we write in detail, with all the brackets and signs:
It seems incredibly difficult to write out so carefully. But it only seems so. Give it a try. Well, or choose. What's better, fast or right? Besides, I will make you happy. After a while, there will be no need to write everything down so carefully. It will work out right on its own. Especially if you use practical techniques that are described below. This evil example with a bunch of minuses can be solved easily and without errors!
But, often, quadratic equations look slightly different. For example, like this:
Did you recognize it?) Yes! This incomplete quadratic equations.
Solving incomplete quadratic equations.
They can also be solved using a general formula. You just need to understand correctly what they are equal to here. a, b and c.
Have you figured it out? In the first example a = 1; b = -4; A c? It's not there at all! Well yes, that's right. In mathematics this means that c = 0 ! That's all. Substitute zero into the formula instead c, and we will succeed. Same with the second example. Only we don’t have zero here With, A b !
But incomplete quadratic equations can be solved much more simply. Without any formulas. Let's consider the first incomplete equation. What can you do on the left side? You can take X out of brackets! Let's take it out.
And what from this? And the fact that the product equals zero if and only if any of the factors equals zero! Don't believe me? Okay, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? That's it...
Therefore, we can confidently write: x 1 = 0, x 2 = 4.
All. These will be the roots of our equation. Both are suitable. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much simpler than using the general formula. Let me note, by the way, which X will be the first and which will be the second - absolutely indifferent. It is convenient to write in order, x 1- what is smaller and x 2- that which is greater.
The second equation can also be solved simply. Move 9 to the right side. We get:
All that remains is to extract the root from 9, and that’s it. It will turn out:
Also two roots . x 1 = -3, x 2 = 3.
This is how all incomplete quadratic equations are solved. Either by placing X out of brackets, or by simply moving the number to the right and then extracting the root.
It is extremely difficult to confuse these techniques. Simply because in the first case you will have to extract the root of X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets...
Discriminant. Discriminant formula.
Magic word discriminant ! Rarely a high school student has not heard this word! The phrase “we solve through a discriminant” inspires confidence and reassurance. Because there is no need to expect tricks from the discriminant! It is simple and trouble-free to use.) I remind you of the most general formula for solving any quadratic equations:
The expression under the root sign is called a discriminant. Typically the discriminant is denoted by the letter D. Discriminant formula:
D = b 2 - 4ac
And what is so remarkable about this expression? Why did it deserve a special name? What the meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically call it anything... Letters and letters.
Here's the thing. When solving a quadratic equation using this formula, it is possible only three cases.
1. The discriminant is positive. This means the root can be extracted from it. Whether the root is extracted well or poorly is another question. What is important is what is extracted in principle. Then your quadratic equation has two roots. Two different solutions.
2. Discriminant equal to zero. Then you will have one solution. Since adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not one root, but two identical. But, in a simplified version, it is customary to talk about one solution.
3. The discriminant is negative. The square root of a negative number cannot be taken. Well, okay. This means there are no solutions.
Honestly speaking, when simple solution quadratic equations, the concept of a discriminant is not particularly required. We substitute the values of the coefficients into the formula and count. Everything happens there by itself, two roots, one, and none. However, when solving more difficult tasks, without knowledge meaning and formula of the discriminant not enough. Especially in equations with parameters. Such equations are aerobatics for the State Examination and the Unified State Examination!)
So, how to solve quadratic equations through the discriminant you remembered. Or you learned, which is also not bad.) You know how to correctly determine a, b and c. Do you know how? attentively substitute them into the root formula and attentively count the result. You understand that the key word here is attentively?
Now take note of practical techniques that dramatically reduce the number of errors. The same ones that are due to inattention... For which it later becomes painful and offensive...
First appointment
. Don’t be lazy before solving a quadratic equation and bring it to standard form. What does this mean?
Let's say that after all the transformations you get the following equation:
Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c. Construct the example correctly. First, X squared, then without square, then the free term. Like this:
And again, don’t rush! A minus in front of an X squared can really upset you. It's easy to forget... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the entire equation by -1. We get:
But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example. Decide for yourself. You should now have roots 2 and -1.
Reception second. Check the roots! According to Vieta's theorem. Don't be scared, I'll explain everything! Checking last thing the equation. Those. the one we used to write down the root formula. If (as in this example) the coefficient a = 1, checking the roots is easy. It is enough to multiply them. The result should be a free member, i.e. in our case -2. Please note, not 2, but -2! Free member with your sign . If it doesn’t work out, it means they’ve already screwed up somewhere. Look for the error.
If it works, you need to add the roots. Last and final check. The coefficient should be b With opposite
familiar. In our case -1+2 = +1. A coefficient b, which is before the X, is equal to -1. So, everything is correct!
It’s a pity that this is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! There will be fewer and fewer errors.
Reception third . If your equation has fractional coefficients, get rid of the fractions! Multiply the equation by common denominator, as described in the lesson "How to solve equations? Identical transformations." When working with fractions, errors keep creeping in for some reason...
By the way, I promised to simplify the evil example with a bunch of minuses. Please! Here he is.
In order not to get confused by the minuses, we multiply the equation by -1. We get:
That's all! Solving is a pleasure!
So, let's summarize the topic.
Practical tips:
1. Before solving, we bring the quadratic equation to standard form and build it Right.
2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying the entire equation by -1.
3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding factor.
4. If x squared is pure, its coefficient equal to one, the solution can be easily verified using Vieta's theorem. Do it!
Now we can decide.)
Solve equations:
8x 2 - 6x + 1 = 0
x 2 + 3x + 8 = 0
x 2 - 4x + 4 = 0
(x+1) 2 + x + 1 = (x+1)(x+2)
Answers (in disarray):
x 1 = 0
x 2 = 5
x 1.2 =2
x 1 = 2
x 2 = -0.5
x - any number
x 1 = -3
x 2 = 3
no solutions
x 1 = 0.25
x 2 = 0.5
Does everything fit? Great! Quadratic equations are not your headache. The first three worked, but the rest didn’t? Then the problem is not with quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.
Doesn't quite work out? Or does it not work out at all? Then Section 555 will help you. All these examples are broken down there. Shown main errors in the solution. Of course, it also talks about the use identity transformations in solving various equations. Helps a lot!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
I hope that after studying this article you will learn how to find the roots of a complete quadratic equation.
Using the discriminant, only complete quadratic equations are solved; to solve incomplete quadratic equations, other methods are used, which you will find in the article “Solving incomplete quadratic equations.”
What quadratic equations are called complete? This equations of the form ax 2 + b x + c = 0, where coefficients a, b and c are not equal to zero. So, to solve a complete quadratic equation, we need to calculate the discriminant D.
D = b 2 – 4ac.
Depending on the value of the discriminant, we will write down the answer.
If the discriminant is a negative number (D< 0),то корней нет.
If the discriminant is zero, then x = (-b)/2a. When the discriminant is a positive number (D > 0),
then x 1 = (-b - √D)/2a, and x 2 = (-b + √D)/2a.
For example. Solve the equation x 2– 4x + 4= 0.
D = 4 2 – 4 4 = 0
x = (- (-4))/2 = 2
Answer: 2.
Solve Equation 2 x 2 + x + 3 = 0.
D = 1 2 – 4 2 3 = – 23
Answer: no roots.
Solve Equation 2 x 2 + 5x – 7 = 0.
D = 5 2 – 4 2 (–7) = 81
x 1 = (-5 - √81)/(2 2)= (-5 - 9)/4= – 3.5
x 2 = (-5 + √81)/(2 2) = (-5 + 9)/4=1
Answer: – 3.5; 1.
So let’s imagine the solution of complete quadratic equations using the diagram in Figure 1.
Using these formulas you can solve any complete quadratic equation. You just need to be careful to the equation was written as a polynomial standard view
A x 2 + bx + c, otherwise you may make a mistake. For example, in writing the equation x + 3 + 2x 2 = 0, you can mistakenly decide that
a = 1, b = 3 and c = 2. Then
D = 3 2 – 4 1 2 = 1 and then the equation has two roots. And this is not true. (See solution to example 2 above).
Therefore, if the equation is not written as a polynomial of the standard form, first the complete quadratic equation must be written as a polynomial of the standard form (the monomial with the largest exponent should come first, that is A x 2 , then with less – bx and then a free member With.
When solving the reduced quadratic equation and a quadratic equation with an even coefficient in the second term, you can use other formulas. Let's get acquainted with these formulas. If in a complete quadratic equation the second term has an even coefficient (b = 2k), then you can solve the equation using the formulas shown in the diagram in Figure 2.
A complete quadratic equation is called reduced if the coefficient at x 2 is equal to one and the equation takes the form x 2 + px + q = 0. Such an equation can be given for solution, or it can be obtained by dividing all coefficients of the equation by the coefficient A, standing at x 2 .
Figure 3 shows a diagram for solving the reduced square 
equations. Let's look at an example of the application of the formulas discussed in this article.
Example. Solve the equation
3x 2 + 6x – 6 = 0.
Let's solve this equation using the formulas shown in the diagram in Figure 1.
D = 6 2 – 4 3 (– 6) = 36 + 72 = 108
√D = √108 = √(36 3) = 6√3
x 1 = (-6 - 6√3)/(2 3) = (6 (-1- √(3)))/6 = –1 – √3
x 2 = (-6 + 6√3)/(2 3) = (6 (-1+ √(3)))/6 = –1 + √3
Answer: –1 – √3; –1 + √3
You can notice that the coefficient of x in this equation even number, that is, b = 6 or b = 2k, whence k = 3. Then let’s try to solve the equation using the formulas given in the diagram of the figure D 1 = 3 2 – 3 · (– 6) = 9 + 18 = 27
√(D 1) = √27 = √(9 3) = 3√3
x 1 = (-3 - 3√3)/3 = (3 (-1 - √(3)))/3 = – 1 – √3
x 2 = (-3 + 3√3)/3 = (3 (-1 + √(3)))/3 = – 1 + √3
Answer: –1 – √3; –1 + √3. Noticing that all the coefficients in this quadratic equation are divisible by 3 and performing the division, we get the reduced quadratic equation x 2 + 2x – 2 = 0 Solve this equation using the formulas for the reduced quadratic 
equations figure 3.
D 2 = 2 2 – 4 (– 2) = 4 + 8 = 12
√(D 2) = √12 = √(4 3) = 2√3
x 1 = (-2 - 2√3)/2 = (2 (-1 - √(3)))/2 = – 1 – √3
x 2 = (-2 + 2√3)/2 = (2 (-1+ √(3)))/2 = – 1 + √3
Answer: –1 – √3; –1 + √3.
As you can see, when solving this equation using different formulas, we received the same answer. Therefore, having thoroughly mastered the formulas shown in the diagram in Figure 1, you will always be able to solve any complete quadratic equation.
website, when copying material in full or in part, a link to the source is required.
Before we learn how to find the discriminant of a quadratic equation of the form ax2+bx+c=0 and how to find the roots of a given equation, we need to remember the definition of a quadratic equation. The equation, which looks like ax 2 + bx + c = 0 (where a, b and c are any numbers, also remember that a ≠ 0) is quadratic. We divide all quadratic equations into three categories:
- those that have no roots;
- there is one root in the equation;
- there are two roots.
In order to determine the number of roots in the equation, we need a discriminant.
How to find a discriminant. Formula
We are given: ax 2 + bx + c = 0.
Discriminant formula: D = b 2 - 4ac.
How to find the roots of a discriminant
The number of roots is determined by the discriminant sign:
- D = 0, the equation has one root;
- D > 0, the equation has two roots.
The roots of a quadratic equation are found using the following formula:
X1= -b + √D/2a; X2= -b + √D/2a.
If D = 0, then you can safely use any of the presented formulas. You will get the same answer either way. And if it turns out that D > 0, then you don’t have to count anything, since the equation has no roots.
It must be said that finding a discriminant is not so difficult if you know the formulas and carefully carry out the calculations. Sometimes errors occur when substituting negative numbers into the formula (you need to remember that minus by minus gives a plus). Be careful and everything will work out!