The right side of the equation 129 64 what is the discriminant. 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 rockets. Using such calculations, the trajectories of movement of the most different bodies, 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 in hiking trips, on sports competitions, in stores when shopping and in other very common situations.
Let's break the expression into its component factors
The degree of the equation is determined maximum value degree of the variable that this 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. Substituting required values, equating right side 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 decide specified tasks and in more complex 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 of an incomplete second-order equation is an expression represented in the language of letters in such a way that the right-hand 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 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
Need for this kind of calculations appeared in ancient times, because the development of mathematics in many ways in those distant times was 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 a form of equation that 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 variable x 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 y 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 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.
Among the entire school algebra curriculum, one of the most extensive topics is the topic of quadratic equations. In this case, a quadratic equation is understood as an equation of the form ax 2 + bx + c = 0, where a ≠ 0 (read: a multiplied by x squared plus be x plus ce is equal to zero, where a is not equal to zero). In this case, the main place is occupied by formulas for finding the discriminant of a quadratic equation of the specified type, which is understood as an expression that allows one to determine the presence or absence of roots of a quadratic equation, as well as their number (if any).
Formula (equation) of the discriminant of a quadratic equation
The generally accepted formula for the discriminant of a quadratic equation is as follows: D = b 2 – 4ac. By calculating the discriminant using the specified formula, you can not only determine the presence and number of roots of a quadratic equation, but also choose a method for finding these roots, of which there are several depending on the type of quadratic equation.
What does it mean if the discriminant is zero \ Formula for the roots of a quadratic equation if the discriminant is zero
The discriminant, as follows from the formula, is denoted by the Latin letter D. In the case when the discriminant is equal to zero, it should be concluded that a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, has only one root, which is calculated by simplified formula. This formula applies only when the discriminant is zero and looks like this: x = –b/2a, where x is the root of the quadratic equation, b and a are the corresponding variables of the quadratic equation. To find the root of a quadratic equation, you need to divide the negative value of the variable b by twice the value of the variable a. The resulting expression will be the solution to a quadratic equation.
Solving a quadratic equation using a discriminant
If, when calculating the discriminant using the above formula, a positive value is obtained (D is greater than zero), then the quadratic equation has two roots, which are calculated using the following formulas: x 1 = (–b + vD)/2a, x 2 = (–b – vD) /2a. Most often, the discriminant is not calculated separately, but the radical expression in the form of the discriminant formula is simply substituted into the value D from which the root is extracted. If the variable b has an even value, then to calculate the roots of a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, you can also use the following formulas: x 1 = (–k + v(k2 – ac))/a , x 2 = (–k + v(k2 – ac))/a, where k = b/2.
In some cases, to practically solve quadratic equations, you can use Vieta’s Theorem, which states that for the sum of the roots of a quadratic equation of the form x 2 + px + q = 0 the value x 1 + x 2 = –p will be true, and for the product of the roots of the specified equation – expression x 1 x x 2 = q.
Can the discriminant be less than zero?
When calculating the discriminant value, you may encounter a situation that does not fall under any of the described cases - when the discriminant has a negative value (that is, less than zero). In this case, it is generally accepted that a quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0, has no real roots, therefore, its solution will be limited to calculating the discriminant, and the above formulas for the roots of a quadratic equation will not apply in this case there will be. At the same time, in the answer to the quadratic equation it is written that “the equation has no real roots.”
Explanatory video:
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 can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and 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.
In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems 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 decimal fractions 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 a numerical fraction, the numerator is separated from the denominator by a division sign: /
The whole part is separated from the fraction by the ampersand sign: &
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
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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 given quadratic equations 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 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.
Let's 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 is 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 from 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 reduced 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\( \begin(array)(l) x_1+x_2=-p \\ x_1 \cdot x_2=q \end(array) \right. \)
A quadratic equation is an equation that looks like ax 2 + dx + c = 0. It has meaning a,c And With any numbers, and A not equal to zero.
All quadratic equations are divided into several types, namely:
Equations with only one root.
-Equations with two different roots.
-Equations in which there are no roots at all.
This distinguishes linear equations in which the root is always the same, from square ones. In order to understand how many roots are in the expression, you need Discriminant of a quadratic equation.
Let's assume our equation ax 2 + dx + c =0. Means discriminant of a quadratic equation -
D = b 2 - 4 ac
And this must be remembered forever. Using this equation we determine the number of roots in the quadratic equation. And we do it this way:
When D is less than zero, there are no roots in the equation.
- When D is zero, there is only one root.
- When D is greater than zero, the equation has two roots.
Remember that the discriminant shows how many roots are in the equation without changing the signs.
Let's consider for clarity:
We need to find out how many roots there are in this quadratic equation.
1) x 2 - 8x + 12 = 0
2)5x 2 + 3x + 7 = 0
3) x 2 -6x + 9 = 0
We enter the values into the first equation and find the discriminant.
a = 1, b = -8, c = 12
D = (-8) 2 - 4 * 1 * 12 = 64 - 48 = 16
The discriminant has a plus sign, which means there are two roots in this equality.
We do the same with the second equation
a = 1, b = 3, c = 7
D = 3 2 - 4 * 5 * 7 = 9 - 140 = - 131
The value is negative, which means there are no roots in this equality.
Let us expand the following equation by analogy.
a = 1, b = -6, c = 9
D = (-6) 2 - 4 * 1 * 9 = 36 - 36 = 0
as a consequence, we have one root in the equation.
It is important that in each equation we wrote out the coefficients. Of course, this is not a very long process, but it helped us not get confused and prevented errors from occurring. If you solve similar equations very often, you will be able to perform the calculations mentally and know in advance how many roots the equation has.
Let's look at another example:
1) x 2 - 2x - 3 = 0
2) 15 - 2x - x 2 = 0
3) x 2 + 12x + 36 = 0
Let's lay out the first
a = 1, b = -2, c = -3
D =(-2) 2 - 4 * 1 * (-3) = 16, which is greater than zero, which means two roots, let’s derive them
x 1 = 2+?16/2 * 1 = 3, x 2 = 2-?16/2 * 1 = -1.
We lay out the second
a = -1, b = -2, c = 15
D = (-2) 2 - 4 * 4 * (-1) * 15 = 64, which is greater than zero and also has two roots. Let's output them:
x 1 = 2+?64/2 * (-1) = -5, x 2 = 2-?64/2 *(-1) = 3.
We lay out the third
a = 1, b = 12, c = 36
D = 12 2 - 4 * 1 * 36 =0, which is equal to zero and has one root
x = -12 + ?0/2 * 1 = -6.
Solving these equations is not difficult.
If we are given an incomplete quadratic equation. Such as
1x 2 + 9x = 0
2x 2 - 16 = 0
These equations differ from those above, since it is not complete, there is no third value in it. But despite this, it is simpler than a complete quadratic equation and there is no need to look for a discriminant in it.
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This topic may seem difficult at first due to many not so simple formulas. Not only do the quadratic equations themselves have long notations, but the roots are also found through the discriminant. In total, three new formulas are obtained. Not very easy to remember. This is possible only after solving such equations frequently. Then all the formulas will be remembered by themselves.
General view of a quadratic equation
Here we propose their explicit recording, when the largest degree is written first, and then in descending order. There are often situations when the terms are inconsistent. Then it is better to rewrite the equation in descending order of the degree of the variable.
Let us introduce some notation. They are presented in the table below.
If we accept these notations, all quadratic equations are reduced to the following notation.
Moreover, the coefficient a ≠ 0. Let this formula be designated number one.
When an equation is given, it is not clear how many roots there will be in the answer. Because one of three options is always possible:
- the solution will have two roots;
- the answer will be one number;
- the equation will have no roots at all.
And until the decision is finalized, it is difficult to understand which option will appear in a particular case.
Types of recordings of quadratic equations
There may be different entries in tasks. They won't always look like general formula quadratic equation. Sometimes it will be missing some terms. What was written above is the complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.
Moreover, only terms with coefficients “b” and “c” can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of equations will be as follows:
So, there are only two types; in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second - three.
Discriminant and dependence of the number of roots on its value
You need to know this number in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have number four.
After substituting the coefficient values into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. At negative number the roots of the quadratic equation will be missing. If it is equal to zero, there will be only one answer.
How to solve a complete quadratic equation?
In fact, consideration of this issue has already begun. Because first you need to find a discriminant. After it is determined that there are roots of the quadratic equation, and their number is known, you need to use formulas for the variables. If there are two roots, then you need to apply the following formula.
Since it contains a “±” sign, there will be two meanings. The expression under the square root sign is the discriminant. Therefore, the formula can be rewritten differently.
Formula number five. From the same record it is clear that if the discriminant is equal to zero, then both roots will take the same values.
If solving quadratic equations has not yet been worked out, then it is better to write down the values of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.
How to solve an incomplete quadratic equation?
Everything is much simpler here. There is not even a need for additional formulas. And those that have already been written down for the discriminant and the unknown will not be needed.
Let's first consider incomplete equation at number two. In this equality, it is necessary to take the unknown quantity out of brackets and solve the linear equation, which will remain in brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a multiplier consisting of the variable itself. The second one will be obtained by solving a linear equation.
The incomplete equation number three is solved by moving the number from the left side of the equality to the right. Then you need to divide by the coefficient facing the unknown. All that remains is to extract the square root and remember to write it down twice with opposite signs.
Below are some actions that will help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings can cause poor grades when studying the extensive topic “Quadratic Equations (8th Grade).” Subsequently, these actions will not need to be performed constantly. Because a stable skill will appear.
- First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without a degree, and last - just a number.
- If a minus appears before the coefficient “a”, it can complicate the work for a beginner studying quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by “-1”. This means that all terms will change sign to the opposite.
- It is recommended to get rid of fractions in the same way. Simply multiply the equation by the appropriate factor so that the denominators cancel out.
Examples
It is required to solve the following quadratic equations:
x 2 − 7x = 0;
15 − 2x − x 2 = 0;
x 2 + 8 + 3x = 0;
12x + x 2 + 36 = 0;
(x+1) 2 + x + 1 = (x+1)(x+2).
The first equation: x 2 − 7x = 0. It is incomplete, so it is solved as described for formula number two.
After taking it out of brackets, it turns out: x (x - 7) = 0.
The first root takes the value: x 1 = 0. The second will be found from linear equation: x - 7 = 0. It is easy to see that x 2 = 7.
Second equation: 5x 2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.
After moving 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be the numbers: x 1 = √6, x 2 = - √6.
The third equation: 15 − 2x − x 2 = 0. Hereinafter, solving quadratic equations will begin with their rewriting in standard view: − x 2 − 2x + 15 = 0. Now it’s time to use the second useful advice and multiply everything by minus one. It turns out x 2 + 2x - 15 = 0. Using the fourth formula, you need to calculate the discriminant: D = 2 2 - 4 * (- 15) = 4 + 60 = 64. It is a positive number. From what is said above, it turns out that the equation has two roots. They need to be calculated using the fifth formula. It turns out that x = (-2 ± √64) / 2 = (-2 ± 8) / 2. Then x 1 = 3, x 2 = - 5.
The fourth equation x 2 + 8 + 3x = 0 is transformed into this: x 2 + 3x + 8 = 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: “There are no roots.”
The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x = -12/ (2 * 1) = -6.
The sixth equation (x+1) 2 + x + 1 = (x+1)(x+2) requires transformations, which consist in the fact that you need to bring similar terms, first opening the brackets. In place of the first there will be the following expression: x 2 + 2x + 1. After the equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x = 0. It has become incomplete . Something similar to this has already been discussed a little higher. The roots of this will be the numbers 0 and 1.
