What are the roots equal to if the discriminant is 0. Quadratic equation
IN modern society the ability to perform operations with equations containing a variable squared can be useful in many areas of activity and is widely used in practice in scientific and technical developments. Evidence of this can be found in the design of sea and river vessels, aircraft and missiles. Using such calculations, the trajectories of movement of a wide variety of bodies are determined, including space objects. Examples with the solution of quadratic equations are used not only in economic forecasting, in the design and construction of buildings, but also in the most ordinary everyday circumstances. They may be needed on hiking trips, at sporting events, in stores when making purchases and in other very common situations.
Let's break the expression into its component factors
The degree of an equation is determined by the maximum value of the degree of the variable that the expression contains. If it is equal to 2, then such an equation is called quadratic.
If we speak in the language of formulas, then the indicated expressions, no matter how they look, can always be brought to the form when the left side of the expression consists of three terms. Among them: ax 2 (that is, a variable squared with its coefficient), bx (an unknown without a square with its coefficient) and c (a free component, that is, an ordinary number). All this on the right side is equal to 0. In the case when such a polynomial lacks one of its constituent terms, with the exception of ax 2, it is called an incomplete quadratic equation. Examples with the solution of such problems, the values of the variables in which are easy to find, should be considered first.
If the expression looks like it has two terms on the right side, more precisely ax 2 and bx, the easiest way to find x is by putting the variable out of brackets. Now our equation will look like this: x(ax+b). Next, it becomes obvious that either x=0, or the problem comes down to finding a variable from the following expression: ax+b=0. This is dictated by one of the properties of multiplication. The rule states that the product of two factors results in 0 only if one of them is zero.
Example
x=0 or 8x - 3 = 0
As a result, we get two roots of the equation: 0 and 0.375.
Equations of this kind can describe the movement of bodies under the influence of gravity, which began to move from a certain point taken as the origin of coordinates. Here the mathematical notation takes the following form: y = v 0 t + gt 2 /2. By substituting the necessary values, equating the right side to 0 and finding possible unknowns, you can find out the time that passes from the moment the body rises to the moment it falls, as well as many other quantities. But we'll talk about this later.
Factoring an Expression
The rule described above makes it possible to solve these problems in more difficult cases. Let's look at examples of solving quadratic equations of this type.
X 2 - 33x + 200 = 0
This quadratic trinomial is complete. First, let's transform the expression and factor it. There are two of them: (x-8) and (x-25) = 0. As a result, we have two roots 8 and 25.
Examples with solving quadratic equations in grade 9 allow this method to find a variable in expressions not only of the second, but even of the third and fourth orders.
For example: 2x 3 + 2x 2 - 18x - 18 = 0. When factoring the right side into factors with a variable, there are three of them, that is, (x+1), (x-3) and (x+3).
As a result, it becomes obvious that this equation has three roots: -3; -1; 3.
Square Root
Another case incomplete equation the second order is an expression represented in the language of letters in such a way that the right side is constructed from the components ax 2 and c. Here, to obtain the value of the variable, the free term is transferred to right side, and after that from both sides of the equality we extract Square root. It should be noted that in this case there are usually two roots of the equation. The only exceptions can be equalities that do not contain a term with at all, where the variable is equal to zero, as well as variants of expressions when the right side turns out to be negative. In the latter case, there are no solutions at all, since the above actions cannot be performed with roots. Examples of solutions to quadratic equations of this type should be considered.
In this case, the roots of the equation will be the numbers -4 and 4.
Calculation of land area
The need for this kind of calculations appeared in ancient times, because the development of mathematics in those distant times was largely determined by the need to determine with the greatest accuracy the areas and perimeters of land plots.
We should also consider examples of solving quadratic equations based on problems of this kind.
So, let's say there is a rectangular plot of land, the length of which is 16 meters greater than the width. You should find the length, width and perimeter of the site if you know that its area is 612 m2.
To get started, let's first create the necessary equation. Let us denote by x the width of the area, then its length will be (x+16). From what has been written it follows that the area is determined by the expression x(x+16), which, according to the conditions of our problem, is 612. This means that x(x+16) = 612.
Solving complete quadratic equations, and this expression is exactly that, cannot be done in the same way. Why? Although the left side still contains two factors, their product does not equal 0 at all, so different methods are used here.
Discriminant
First of all, let's make the necessary transformations, then appearance of this expression will look like this: x 2 + 16x - 612 = 0. This means that we have received an expression in a form corresponding to the previously specified standard, where a=1, b=16, c=-612.
This could be an example of solving quadratic equations using a discriminant. Here necessary calculations are produced according to the scheme: D = b 2 - 4ac. This auxiliary quantity not only makes it possible to find the required quantities in a second-order equation, it determines the quantity possible options. If D>0, there are two of them; for D=0 there is one root. In case D<0, никаких шансов для решения у уравнения вообще не имеется.
About roots and their formula
In our case, the discriminant is equal to: 256 - 4(-612) = 2704. This suggests that our problem has an answer. If you know k, the solution of quadratic equations must be continued using the formula below. It allows you to calculate the roots.
This means that in the presented case: x 1 =18, x 2 =-34. The second option in this dilemma cannot be a solution, because the dimensions of the land plot cannot be measured in negative quantities, which means x (that is, the width of the plot) is 18 m. From here we calculate the length: 18+16=34, and the perimeter 2(34+ 18)=104(m2).
Examples and tasks
We continue our study of quadratic equations. Examples and detailed solutions of several of them will be given below.
1) 15x 2 + 20x + 5 = 12x 2 + 27x + 1
Let's move everything to left side equality, we will make a transformation, that is, we will obtain the form of the equation, which is usually called standard, and we will equate it to zero.
15x 2 + 20x + 5 - 12x 2 - 27x - 1 = 0
Adding similar ones, we determine the discriminant: D = 49 - 48 = 1. This means our equation will have two roots. Let's calculate them according to the above formula, which means that the first of them will be equal to 4/3, and the second to 1.
2) Now let's solve mysteries of a different kind.
Let's find out if there are any roots here x 2 - 4x + 5 = 1? To obtain a comprehensive answer, let’s reduce the polynomial to the corresponding usual form and calculate the discriminant. In the above example, it is not necessary to solve the quadratic equation, because this is not the essence of the problem at all. In this case, D = 16 - 20 = -4, which means there really are no roots.
Vieta's theorem
It is convenient to solve quadratic equations using the above formulas and the discriminant, when the square root is taken from the value of the latter. But this does not always happen. However, there are many ways to obtain the values of variables in this case. Example: solving quadratic equations using Vieta's theorem. She is named after who lived in the 16th century in France and made a brilliant career thanks to his mathematical talent and connections at court. His portrait can be seen in the article.
The pattern that the famous Frenchman noticed was as follows. He proved that the roots of the equation add up numerically to -p=b/a, and their product corresponds to q=c/a.
Now let's look at specific tasks.
3x 2 + 21x - 54 = 0
For simplicity, let's transform the expression:
x 2 + 7x - 18 = 0
Let's use Vieta's theorem, this will give us the following: the sum of the roots is -7, and their product is -18. From here we get that the roots of the equation are the numbers -9 and 2. After checking, we will make sure that these variable values really fit into the expression.
Parabola graph and equation
The concepts of quadratic function and quadratic equations are closely related. Examples of this have already been given earlier. Now let's look at some mathematical riddles in a little more detail. Any equation of the described type can be represented visually. Such a relationship, drawn as a graph, is called a parabola. Its various types are presented in the figure below.
Any parabola has a vertex, that is, a point from which its branches emerge. If a>0, they go high to infinity, and when a<0, они рисуются вниз. Простейшим примером подобной зависимости является функция y = x 2 . В данном случае в уравнении x 2 =0 неизвестное может принимать только одно значение, то есть х=0, а значит существует только один корень. Это неудивительно, ведь здесь D=0, потому что a=1, b=0, c=0. Выходит формула корней (точнее одного корня) квадратного уравнения запишется так: x = -b/2a.
Visual representations of functions help solve any equations, including quadratic ones. This method is called graphical. And the value of the x variable is the abscissa coordinate at the points where the graph line intersects with 0x. The coordinates of the vertex can be found using the formula just given x 0 = -b/2a. And by substituting the resulting value into the original equation of the function, you can find out y 0, that is, the second coordinate of the vertex of the parabola, which belongs to the ordinate axis.
The intersection of the branches of a parabola with the abscissa axis
There are a lot of examples of solving quadratic equations, but there are also general patterns. Let's look at them. It is clear that the intersection of the graph with the 0x axis for a>0 is possible only if 0 takes negative values. And for a<0 координата у 0 должна быть положительна. Для указанных вариантов D>0. Otherwise D<0. А когда D=0, вершина параболы расположена непосредственно на оси 0х.
From the graph of the parabola you can also determine the roots. The opposite is also true. That is, if it is not easy to obtain a visual representation of a quadratic function, you can equate the right side of the expression to 0 and solve the resulting equation. And knowing the points of intersection with the 0x axis, it is easier to construct a graph.
From the history
Using equations containing a squared variable, in the old days they not only made mathematical calculations and determined the areas of geometric figures. The ancients needed such calculations for grand discoveries in the fields of physics and astronomy, as well as for making astrological forecasts.
As modern scientists suggest, the inhabitants of Babylon were among the first to solve quadratic equations. This happened four centuries before our era. Of course, their calculations were radically different from those currently accepted and turned out to be much more primitive. For example, Mesopotamian mathematicians had no idea about the existence of negative numbers. They were also unfamiliar with other subtleties that any modern schoolchild knows.
Perhaps even earlier than the scientists of Babylon, the sage from India Baudhayama began solving quadratic equations. This happened about eight centuries before the era of Christ. True, the second-order equations, the methods for solving which he gave, were the simplest. Besides him, Chinese mathematicians were also interested in similar questions in the old days. In Europe, quadratic equations began to be solved only at the beginning of the 13th century, but later they were used in their works by such great scientists as Newton, Descartes and many others.
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times, and since then their use has only increased. The discriminant allows you to solve any quadratic equation using a general formula, which has the following form:
The discriminant formula depends on the degree of the polynomial. The above formula is suitable for solving quadratic equations of the following form:
The discriminant has the following properties that you need to know:
* "D" is 0 when the polynomial has multiple roots (equal roots);
* "D" is a symmetric polynomial with respect to the roots of the polynomial and is therefore a polynomial in its coefficients; moreover, the coefficients of this polynomial are integers regardless of the extension in which the roots are taken.
Let's say we are given a quadratic equation of the following form:
1 equation
According to the formula we have:
Since \, the equation has 2 roots. Let's define them:
Where can I solve an equation using a discriminant online solver?
You can solve the equation on our website https://site. The free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch the video instructions and find out how to solve the equation on our website. And if you have any questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.
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 the 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 the equation: \(x_(1)=\)\(\ frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b-\sqrt(D))(2a)\). Let's look at each case in more detail .
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 : \(x=2\)
Discriminant is a multi-valued term. In this article we will talk about the discriminant of a polynomial, which allows you to determine whether a given polynomial has valid solutions. The formula for the quadratic polynomial is found in the school course on algebra and analysis. How to find a discriminant? What is needed to solve the equation?
A quadratic polynomial or equation of the second degree is called i * w ^ 2 + j * w + k equals 0, where “i” and “j” are the first and second coefficients, respectively, “k” is a constant, sometimes called the “dismissive term,” and “w” is a variable. Its roots will be all the values of the variable at which it turns into an identity. Such an equality can be rewritten as the product of i, (w - w1) and (w - w2) equal to 0. In this case, it is obvious that if the coefficient “i” does not become zero, then the function on the left side will become zero only if if x takes the value w1 or w2. These values are the result of setting the polynomial equal to zero.
To find the value of a variable at which a quadratic polynomial vanishes, an auxiliary construction is used, built on its coefficients and called a discriminant. This design is calculated according to the formula D equals j * j - 4 * i * k. Why is it used?
- It tells whether there are valid results.
- She helps calculate them.
How does this value show the presence of real roots:
- If it is positive, then two roots can be found in the region of real numbers.
- If the discriminant is zero, then both solutions are the same. We can say that there is only one solution, and it is from the field of real numbers.
- If the discriminant is less than zero, then the polynomial has no real roots.
Calculation options for securing material
For the sum (7 * w^2; 3 * w; 1) equal to 0 We calculate D using the formula 3 * 3 - 4 * 7 * 1 = 9 - 28, we get -19. A discriminant value below zero indicates that there are no results on the actual line.
If we consider 2 * w^2 - 3 * w + 1 equivalent to 0, then D is calculated as (-3) squared minus the product of the numbers (4; 2; 1) and equals 9 - 8, that is, 1. A positive value indicates two results on the real line.
If we take the sum (w ^ 2; 2 * w; 1) and equate it to 0, D is calculated as two squared minus the product of the numbers (4; 1; 1). This expression will simplify to 4 - 4 and go to zero. It turns out that the results are the same. If you look closely at this formula, it will become clear that this is a “complete square”. This means that the equality can be rewritten in the form (w + 1) ^ 2 = 0. It became obvious that the result in this problem is “-1”. In a situation where D is equal to 0, the left side of the equality can always be collapsed using the “square of the sum” formula.
Using a discriminant in calculating roots
This auxiliary construction not only shows the number of real solutions, but also helps to find them. The general calculation formula for a second degree equation is:
w = (-j +/- d) / (2 * i), where d is the discriminant to the power of 1/2.
Let's say the discriminant is below zero, then d is imaginary and the results are imaginary.
D is zero, then d equal to D to the power of 1/2 is also zero. Solution: -j / (2 * i). Again considering 1 * w ^ 2 + 2 * w + 1 = 0, we find results equivalent to -2 / (2 * 1) = -1.
Suppose D > 0, then d is a real number, and the answer here breaks down into two parts: w1 = (-j + d) / (2 * i) and w2 = (-j - d) / (2 * i) . Both results will be valid. Let's look at 2 * w ^ 2 - 3 * w + 1 = 0. Here the discriminant and d are ones. It turns out that w1 is equal to (3 + 1) divided by (2 * 2) or 1, and w2 is equal to (3 - 1) divided by 2 * 2 or 1/2.
The result of equating a quadratic expression to zero is calculated according to the algorithm:
- Determination of quantity valid solutions.
- Calculation d = D^(1/2).
- Finding the result according to the formula (-j +/- d) / (2 * i).
- Substituting the obtained result into the original equality for verification.
Some special cases
Depending on the coefficients, the solution may be somewhat simplified. Obviously, if the coefficient of a variable to the second power is zero, then a linear equality is obtained. When the coefficient of a variable to the first power is zero, then two options are possible:
- the polynomial is expanded into a difference of squares when the free term is negative;
- for a positive constant, no real solutions can be found.
If the free term is zero, then the roots will be (0; -j)
But there are other special cases that simplify finding a solution.
Reduced second degree equation
The given is called such a quadratic trinomial, where the coefficient of the leading term is one. For this situation, Vieta’s theorem is applicable, which states that the sum of the roots is equal to the coefficient of the variable to the first power, multiplied by -1, and the product corresponds to the constant “k”.
Therefore, w1 + w2 equals -j and w1 * w2 equals k if the first coefficient is one. To verify the correctness of this representation, you can express w2 = -j - w1 from the first formula and substitute it into the second equality w1 * (-j - w1) = k. The result is the original equality w1 ^ 2 + j * w1 + k = 0.
It is important to note, that i * w ^ 2 + j * w + k = 0 can be achieved by dividing by “i”. The result will be: w^2 + j1 * w + k1 = 0, where j1 is equal to j/i and k1 is equal to k/i.
Let's look at the already solved 2 * w^2 - 3 * w + 1 = 0 with the results w1 = 1 and w2 = 1/2. We need to divide it in half, as a result w ^ 2 - 3/2 * w + 1/2 = 0. Let's check that the conditions of the theorem are true for the results found: 1 + 1/2 = 3/2 and 1*1/2 = 1 /2.
Even second factor
If the factor of a variable to the first power (j) is divisible by 2, then it will be possible to simplify the formula and look for a solution through a quarter of the discriminant D/4 = (j / 2) ^ 2 - i * k. it turns out w = (-j +/- d/2) / i, where d/2 = D/4 to the power of 1/2.
If i = 1, and the coefficient j is even, then the solution will be the product of -1 and half the coefficient of the variable w, plus/minus the root of the square of this half minus the constant “k”. Formula: w = -j/2 +/- (j^2/4 - k)^1/2.
Higher discriminant order
The discriminant of the second degree trinomial discussed above is the most commonly used special case. In the general case, the discriminant of a polynomial is multiplied squares of the differences of the roots of this polynomial. Therefore, a discriminant equal to zero indicates the presence of at least two multiple solutions.
Consider i * w^3 + j * w^2 + k * w + m = 0.
D = j^2 * k^2 - 4 * i * k^3 - 4 * i^3 * k - 27 * i^2 * m^2 + 18 * i * j * k * m.
Suppose the discriminant exceeds zero. This means that there are three roots in the region of real numbers. At zero there are multiple solutions. If D< 0, то два корня комплексно-сопряженные, которые дают отрицательное значение при возведении в квадрат, а также один корень — вещественный.
Video
Our video will tell you in detail about calculating the discriminant.
Didn't get an answer to your question? Suggest a topic to the authors.
With this math program you can solve quadratic equation.
The program not only gives the answer to the problem, but also displays the solution process in two ways:
- using a discriminant
- using Vieta's theorem (if possible).
Moreover, the answer is displayed as exact, not approximate.
For example, for the equation \(81x^2-16x-1=0\) the answer is displayed in the following form:
This program may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.
This way you can conduct your own training and/or training of yours. younger brothers or sisters, while the level of education in the field of problems being solved increases.
If you are not familiar with the rules for entering a quadratic polynomial, we recommend that you familiarize yourself with them.
Rules for entering a quadratic polynomial
Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q\), etc.
Numbers can be entered as whole or fractional numbers.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.
Rules for entering decimal fractions.
In decimal fractions, the fractional part can be separated from the whole part by either a period or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2
Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.
The denominator cannot be negative.
When entering numerical fraction The numerator is separated from the denominator by a division sign: /
Whole part separated from the fraction by an ampersand: &
Input: 3&1/3 - 5&6/5z +1/7z^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) z + \frac(1)(7)z^2\)
When entering an expression you can use parentheses. In this case, when solving a quadratic equation, the introduced expression is first simplified.
For example: 1/2(y-1)(y+1)-(5y-10&1/2)
Decide
It was discovered that some scripts necessary to solve this problem were not loaded, and the program may not work.
You may have AdBlock enabled.
In this case, disable it and refresh the page.
For the solution to appear, you need to enable JavaScript.
Here are instructions on how to enable JavaScript in your browser.
Because There are a lot of people willing to solve the problem, your request has been queued.
In a few seconds the solution will appear below.
Please wait sec...
If you noticed an error in the solution, then you can write about this in the Feedback Form.
Do not forget indicate which task you decide what enter in the fields.
Our games, puzzles, emulators:
A little theory.
Quadratic equation and its roots. Incomplete quadratic equations
Each of the equations
\(-x^2+6x+1.4=0, \quad 8x^2-7x=0, \quad x^2-\frac(4)(9)=0 \)
looks like
\(ax^2+bx+c=0, \)
where x is a variable, a, b and c are numbers.
In the first equation a = -1, b = 6 and c = 1.4, in the second a = 8, b = -7 and c = 0, in the third a = 1, b = 0 and c = 4/9. Such equations are called quadratic equations.
Definition.
Quadratic equation is called an equation of the form ax 2 +bx+c=0, where x is a variable, a, b and c are some numbers, and \(a \neq 0 \).
The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, the number b is the second coefficient, and the number c is the free term.
In each of the equations of the form ax 2 +bx+c=0, where \(a\neq 0\), the largest power of the variable x is a square. Hence the name: quadratic equation.
Note that a quadratic equation is also called an equation of the second degree, since its left side is a polynomial of the second degree.
Quadratic equation, in which the coefficient of x 2 is equal to 1 is called given quadratic equation. For example, the quadratic equations given are the equations
\(x^2-11x+30=0, \quad x^2-6x=0, \quad x^2-8=0 \)
If in a quadratic equation ax 2 +bx+c=0 at least one of the coefficients b or c is equal to zero, then such an equation is called incomplete quadratic equation. Thus, the equations -2x 2 +7=0, 3x 2 -10x=0, -4x 2 =0 are incomplete quadratic equations. In the first of them b=0, in the second c=0, in the third b=0 and c=0.
There are three types of incomplete quadratic equations:
1) ax 2 +c=0, where \(c \neq 0 \);
2) ax 2 +bx=0, where \(b \neq 0 \);
3) ax 2 =0.
Let's consider solving equations of each of these types.
To solve an incomplete quadratic equation of the form ax 2 +c=0 for \(c \neq 0 \), move its free term to the right side and divide both sides of the equation by a:
\(x^2 = -\frac(c)(a) \Rightarrow x_(1,2) = \pm \sqrt( -\frac(c)(a)) \)
Since \(c \neq 0 \), then \(-\frac(c)(a) \neq 0 \)
If \(-\frac(c)(a)>0\), then the equation has two roots.
If \(-\frac(c)(a) To solve an incomplete quadratic equation of the form ax 2 +bx=0 with \(b \neq 0 \) factor its left side and obtain the equation
\(x(ax+b)=0 \Rightarrow \left\( \begin(array)(l) x=0 \\ ax+b=0 \end(array) \right. \Rightarrow \left\( \begin (array)(l) x=0 \\ x=-\frac(b)(a) \end(array) \right. \)
This means that an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) always has two roots.
An incomplete quadratic equation of the form ax 2 =0 is equivalent to the equation x 2 =0 and therefore has a single root 0.
Formula for the roots of a quadratic equation
Let us now consider how to solve quadratic equations in which both the coefficients of the unknowns and the free term are nonzero.
Let's solve the quadratic equation in general view and as a result we get the formula for the roots. This formula can then be used to solve any quadratic equation.
Solve the quadratic equation ax 2 +bx+c=0
Dividing both sides by a, we obtain the equivalent reduced quadratic equation
\(x^2+\frac(b)(a)x +\frac(c)(a)=0 \)
Let's transform this equation by selecting the square of the binomial:
\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2- \left(\frac(b)(2a)\right)^ 2 + \frac(c)(a) = 0 \Rightarrow \)
The radical expression is called discriminant of a quadratic equation ax 2 +bx+c=0 (“discriminant” in Latin - discriminator). It is designated by the letter D, i.e.
\(D = b^2-4ac\)
Now, using the discriminant notation, we rewrite the formula for the roots of the quadratic equation:
\(x_(1,2) = \frac( -b \pm \sqrt(D) )(2a) \), where \(D= b^2-4ac \)
It's obvious that:
1) If D>0, then the quadratic equation has two roots.
2) If D=0, then the quadratic equation has one root \(x=-\frac(b)(2a)\).
3) If D Thus, depending on the value of the discriminant, a quadratic equation can have two roots (for D > 0), one root (for D = 0) or have no roots (for D When solving a quadratic equation using this formula, it is advisable to do the following way:
1) calculate the discriminant and compare it with zero;
2) if the discriminant is positive or equal to zero, then use the root formula; if the discriminant is negative, then write down that there are no roots.
Vieta's theorem
The given quadratic equation ax 2 -7x+10=0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. We see that the sum of the roots is equal to the second coefficient taken with the opposite sign, and the product of the roots is equal to the free term. Any reduced quadratic equation that has roots has this property.
The sum of the roots of the above quadratic equation is equal to the second coefficient taken with the opposite sign, and the product of the roots is equal to the free term.
Those. Vieta's theorem states that the roots x 1 and x 2 of the reduced quadratic equation x 2 +px+q=0 have the property:
\(\left\( \begin(array)(l) x_1+x_2=-p \\ x_1 \cdot x_2=q \end(array) \right. \)