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 per 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 want 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:
By on this occasion, when the discriminant is zero, the school course says that the result is one root, 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 negative number is not extracted, 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 left and right side divide the equation by 2, that is, simplify it. The 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 discriminants.
Case 1. Coefficient b = 0.
The equation becomes:
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 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+ c =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+ c =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” 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 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 because after solving quadratic equation the resulting roots can be checked in the usual way (through a discriminant). 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 the roots of the equation can be easily found 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 reroll the 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 boil 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 him to standard view(so as not to get confused when deciding).
2. Remember that x is an unknown quantity and it can be denoted by any other letter - t, q, p, h and others.
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.
website, when copying material in full or in part, a link to the source is required.
For example, for the trinomial \(3x^2+2x-7\), the discriminant will be equal to \(2^2-4\cdot3\cdot(-7)=4+84=88\). And for the trinomial \(x^2-5x+11\), it will be equal to \((-5)^2-4\cdot1\cdot11=25-44=-19\).
The discriminant is denoted by \(D\) and is often used in solving. Also, by the value of the discriminant, you can understand what the graph approximately looks like (see below).
Discriminant and roots of a quadratic equation
The discriminant value shows the number of quadratic equations:
- if \(D\) is positive, the equation will have two roots;
- if \(D\) is equal to zero – there is only one root;
- if \(D\) is negative, there are no roots.
This does not need to be taught, it is not difficult to come to such a conclusion, simply knowing that from the discriminant (that is, \(\sqrt(D)\) is included in the formula for calculating the roots of a quadratic equation: \(x_(1)=\)\( \frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b-\sqrt(D))(2a)\) Let's look at each case more details.
If the discriminant is positive
In this case, the root of it is some positive number, which means \(x_(1)\) and \(x_(2)\) will have different meanings, because in the first formula \(\sqrt(D)\) is added , and in the second it is subtracted. And we have two different roots.
Example
: Find the roots of the equation \(x^2+2x-3=0\)
Solution
:
Answer : \(x_(1)=1\); \(x_(2)=-3\)
If the discriminant is zero
How many roots will there be if the discriminant is zero? Let's reason.
The root formulas look like this: \(x_(1)=\)\(\frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b- \sqrt(D))(2a)\) . And if the discriminant is zero, then its root is also zero. Then it turns out:
\(x_(1)=\)\(\frac(-b+\sqrt(D))(2a)\) \(=\)\(\frac(-b+\sqrt(0))(2a)\) \(=\)\(\frac(-b+0)(2a)\) \(=\)\(\frac(-b)(2a)\)
\(x_(2)=\)\(\frac(-b-\sqrt(D))(2a)\) \(=\)\(\frac(-b-\sqrt(0))(2a) \) \(=\)\(\frac(-b-0)(2a)\) \(=\)\(\frac(-b)(2a)\)
That is, the values of the roots of the equation will be the same, because adding or subtracting zero does not change anything.
Example
: Find the roots of the equation \(x^2-4x+4=0\)
Solution
:
|
\(x^2-4x+4=0\) |
We write out the coefficients: |
|
|
\(a=1;\) \(b=-4;\) \(c=4;\) |
We calculate the discriminant using the formula \(D=b^2-4ac\) |
|
|
\(D=(-4)^2-4\cdot1\cdot4=\) |
Finding the roots of the equation |
|
|
\(x_(1)=\) \(\frac(-(-4)+\sqrt(0))(2\cdot1)\)\(=\)\(\frac(4)(2)\) \(=2\) \(x_(2)=\) \(\frac(-(-4)-\sqrt(0))(2\cdot1)\)\(=\)\(\frac(4)(2)\) \(=2\) |
|
We got two identical roots, so there is no point in writing them separately - we write them as one. |
Answer : \(x=2\)
Quadratic equation problems are also studied in school curriculum and in universities. They mean equations of the form a*x^2 + b*x + c = 0, where x- variable, a, b, c – constants; a<>0 . The task is to find the roots of the equation.
Geometric meaning of quadratic equation
The graph of a function that is represented by a quadratic equation is a parabola. The solutions (roots) of a quadratic equation are the points of intersection of the parabola with the abscissa (x) axis. It follows that there are three possible cases:
1) the parabola has no points of intersection with the abscissa axis. This means that it is in the upper plane with branches up or the bottom with branches down. In such cases, the quadratic equation has no real roots (it has two complex roots).
2) the parabola has one point of intersection with the Ox axis. Such a point is called the vertex of the parabola, and the quadratic equation at it acquires its minimum or maximum value. In this case, the quadratic equation has one real root (or two identical roots).
3) The last case is more interesting in practice - there are two points of intersection of the parabola with the abscissa axis. This means that there are two real roots of the equation.
Based on the analysis of the coefficients of the powers of the variables, interesting conclusions can be drawn about the placement of the parabola.
1) If the coefficient a is greater than zero, then the parabola’s branches are directed upward; if it is negative, the parabola’s branches are directed downward.
2) If the coefficient b is greater than zero, then the vertex of the parabola lies in the left half-plane, if it takes a negative value, then in the right.
Derivation of the formula for solving a quadratic equation
Let's transfer the constant from the quadratic equation 
for the equal sign, we get the expression
Multiply both sides by 4a 
To get left perfect square add b^2 to both sides and carry out the transformation
From here we find
Formula for the discriminant and roots of a quadratic equation
The discriminant is the value of the radical expression. If it is positive, then the equation has two real roots, calculated by the formula 
When the discriminant is zero, the quadratic equation has one solution (two coinciding roots), which can be easily obtained from the above formula for D = 0. negative discriminant there are no real root equations. However, solutions to the quadratic equation are found in the complex plane, and their value is calculated using the formula 
Vieta's theorem
Let's consider two roots of a quadratic equation and construct a quadratic equation on their basis. Vieta's theorem itself easily follows from the notation: if we have a quadratic equation of the form 
then the sum of its roots is equal to the coefficient p taken from opposite sign, and the product of the roots of the equation is equal to the free term q. The formulaic representation of the above will look like If in a classical equation the constant a is nonzero, then you need to divide the entire equation by it, and then apply Vieta’s theorem.
Factoring quadratic equation schedule
Let the task be set: factor a quadratic equation. To do this, we first solve the equation (find the roots). Next, we substitute the found roots into the expansion formula for the quadratic equation. This will solve the problem.
Quadratic equation problems
Task 1. Find the roots of a quadratic equation
x^2-26x+120=0 .
Solution: Write down the coefficients and substitute them into the discriminant formula
The root of this value is 14, it is easy to find with a calculator, or remember with frequent use, however, for convenience, at the end of the article I will give you a list of squares of numbers that can often be encountered in such problems.
We substitute the found value into the root formula 
and we get 
Task 2. Solve the equation
2x 2 +x-3=0.
Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant 
By known formulas finding the roots of a quadratic equation 
Task 3. Solve the equation
9x 2 -12x+4=0.
Solution: We have a complete quadratic equation. Determining the discriminant
We got a case where the roots coincide. Find the values of the roots using the formula 
Task 4. Solve the equation
x^2+x-6=0 .
Solution: In cases where there are small coefficients for x, it is advisable to apply Vieta’s theorem. By its condition we obtain two equations
From the second condition we find that the product must be equal to -6. This means that one of the roots is negative. We have the following possible pair of solutions (-3;2), (3;-2) . Taking into account the first condition, we reject the second pair of solutions.
The roots of the equation are equal
Problem 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and its area is 77 cm 2.
Solution: Half the perimeter of a rectangle is equal to the sum of its adjacent sides. Let's denote x as the larger side, then 18-x is its smaller side. The area of the rectangle is equal to the product of these lengths:
x(18-x)=77;
or
x 2 -18x+77=0.
Let's find the discriminant of the equation
Calculating the roots of the equation 
If x=11, That 18's=7 , the opposite is also true (if x=7, then 21's=9).
Problem 6. Factor the quadratic equation 10x 2 -11x+3=0.
Solution: Let's calculate the roots of the equation, to do this we find the discriminant 
We substitute the found value into the root formula and calculate 
We apply the formula for decomposing a quadratic equation by roots 
Opening the brackets we obtain an identity.
Quadratic equation with parameter
Example 1. At what parameter values A , does the equation (a-3)x 2 + (3-a)x-1/4=0 have one root?
Solution: By direct substitution of the value a=3 we see that it has no solution. Next, we will use the fact that with a zero discriminant the equation has one root of multiplicity 2. Let's write out the discriminant 
Let's simplify it and equate it to zero
We have obtained a quadratic equation with respect to the parameter a, the solution of which can be easily obtained using Vieta’s theorem. The sum of the roots is 7, and their product is 12. By simple search we establish that the numbers 3,4 will be the roots of the equation. Since we already rejected the solution a=3 at the beginning of the calculations, the only correct one will be - a=4. Thus, for a=4 the equation has one root.
Example 2. At what parameter values A , the equation a(a+3)x^2+(2a+6)x-3a-9=0 has more than one root?
Solution: Let's first consider the singular points, they will be the values a=0 and a=-3. When a=0, the equation will be simplified to the form 6x-9=0; x=3/2 and there will be one root. For a= -3 we obtain the identity 0=0.
Let's calculate the discriminant 
and find the value of a at which it is positive 
From the first condition we get a>3. For the second, we find the discriminant and roots of the equation 
Let us determine the intervals where the function takes positive values. By substituting the point a=0 we get 3>0
.
So, outside the interval (-3;1/3) the function is negative. Don't forget the point a=0, which should be excluded because it original equation has one root.
As a result, we obtain two intervals that satisfy the conditions of the problem 
There will be many similar tasks in practice, try to figure out the tasks yourself and do not forget to take into account the conditions that are mutually exclusive. Study well the formulas for solving quadratic equations; they are often needed in calculations in various problems and sciences.
The discriminant, like quadratic equations, begins to be studied in an algebra course in the 8th grade. You can solve a quadratic equation through a discriminant and using Vieta's theorem. The method of studying quadratic equations, as well as discriminant formulas, is rather unsuccessfully taught to schoolchildren, like many things in real education. Therefore they pass school years, education in grades 9-11 replaces " higher education"and everyone is looking again - “How to solve a quadratic equation?”, “How to find the roots of the equation?”, “How to find the discriminant?” And...
Discriminant formula
The discriminant D of the quadratic equation a*x^2+bx+c=0 is equal to D=b^2–4*a*c.
The roots (solutions) of a quadratic equation depend on the sign of the discriminant (D):
D>0 – the equation has 2 different real roots;
D=0 - the equation has 1 root (2 matching roots):
D<0
– не имеет действительных корней (в школьной теории). В ВУЗах изучают комплексные числа и уже на множестве комплексных чисел уравнение с отрицательным дискриминантом имеет два комплексных корня.
The formula for calculating the discriminant is quite simple, so many websites offer an online discriminant calculator. We haven’t figured out this kind of scripts yet, so if anyone knows how to implement this, please write to us by email This email address is being protected from spambots. You must have JavaScript enabled to view it. .
General formula for finding the roots of a quadratic equation:
We find the roots of the equation using the formula 
If the coefficient of a squared variable is paired, then it is advisable to calculate not the discriminant, but its fourth part
In such cases, the roots of the equation are found using the formula 
The second way to find roots is Vieta's Theorem.
The theorem is formulated not only for quadratic equations, but also for polynomials. You can read this on Wikipedia or other electronic resources. However, to simplify, let’s consider the part that concerns the above quadratic equations, that is, equations of the form (a=1)
The essence of Vieta's formulas is that the sum of the roots of the equation is equal to the coefficient of the variable, taken with the opposite sign. The product of the roots of the equation is equal to the free term. Vieta's theorem can be written in formulas.
The derivation of Vieta's formula is quite simple. Let's write the quadratic equation through simple factors 
As you can see, everything ingenious is simple at the same time. It is effective to use Vieta’s formula when the difference in modulus of the roots or the difference in the moduli of the roots is 1, 2. For example, the following equations, according to Vieta’s theorem, have roots
Up to equation 4, the analysis should look like this. The product of the roots of the equation is 6, therefore the roots can be the values (1, 6) and (2, 3) or pairs with opposite signs. The sum of the roots is 7 (the coefficient of the variable with the opposite sign). From here we conclude that the solutions to the quadratic equation are x=2; x=3.
It is easier to select the roots of the equation among the divisors of the free term, adjusting their sign in order to fulfill the Vieta formulas. At first, this seems difficult to do, but with practice on a number of quadratic equations, this technique will turn out to be more effective than calculating the discriminant and finding the roots of the quadratic equation in the classical way.
As you can see, the school theory of studying the discriminant and methods of finding solutions to the equation is devoid of practical meaning - “Why do schoolchildren need a quadratic equation?”, “What is the physical meaning of the discriminant?”
Let's try to figure it out What does the discriminant describe?
In the algebra course they study functions, schemes for studying functions and constructing a graph of functions. Of all the functions, the parabola occupies an important place, the equation of which can be written in the form 
So the physical meaning of the quadratic equation is the zeros of the parabola, that is, the points of intersection of the graph of the function with the abscissa axis Ox
I ask you to remember the properties of parabolas that are described below. The time will come to take exams, tests, or entrance exams and you will be grateful for the reference material. The sign of the squared variable corresponds to whether the branches of the parabola on the graph will go up (a>0),
or a parabola with branches down (a<0)
.
The vertex of the parabola lies midway between the roots
Physical meaning of the discriminant:
If the discriminant is greater than zero (D>0) the parabola has two points of intersection with the Ox axis. 
If the discriminant is zero (D=0) then the parabola at the vertex touches the x-axis. 
And the last case, when the discriminant is less than zero (D<0)
– график параболы принадлежит плоскости над осью абсцисс (ветки параболы вверх), или график полностью под осью абсцисс (ветки параболы опущены вниз).
