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How do you write a quadratic equation in the standard form with given solutions -2, 7?

Last updated date: 22nd Jun 2024
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Hint: The quadratic equation is a form of algebraic equation, it is defined as a combination of both variables and constants. Here in this question, we have to find the quadratic equation where here the roots are given. By following the sum product rule, we are going to find the quadratic equation.

The quadratic equation is an equation which contains both variables and constants. By solving the quadratic equation we get two roots for the equation. So we can say that for the quadratic equation has the highest degree 2.
To find the quadratic equation we use the sum product rule. In general the quadratic equation is defined as $a{x^2} + bx + c$ , where a, b and c are constants and x is variable. According to the sum product rule the product of a and c is written as the sum of the b.
Now consider the given roots as $\alpha$ and $\beta$ . So we have $\alpha = - 2$ and $\beta = 7$
In generally, considering the roots the sum product rule of an equation is written as
${x^2} - \left( {\alpha + \beta } \right)x + (\alpha \beta )$ ----------- (1)
The sum of the roots are $\left( {\alpha + \beta } \right) = - 2 + 7$
So we have $\left( {\alpha + \beta } \right) = 5$ ------ (2)
The product of the roots are $\left( {\alpha \beta } \right) = ( - 2)(7)$
So we have $\left( {\alpha \beta } \right) = - 14$ ----------- (3)
Substituting the equation (3) and equation (2) in the equation (1). We get
$\Rightarrow {x^2} - \left( 5 \right)x + ( - 14)$
On simplifying we get
$\Rightarrow {x^2} - 5x - 14$ ------ (4)
Hence we have obtained the quadratic equation for the solutions -2 and 7.
We can also verify the obtained quadratic equation is correct or not by using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ . On comparing the equation (4) to the general form of equation we have a = 1, b = -5 and c = -14.
On substituting these values in the formula we have
$x = \dfrac{{ - ( - 5) \pm \sqrt {{{( - 5)}^2} - 4(1)( - 14)} }}{{2(1)}}$
$\Rightarrow x = \dfrac{{5 \pm \sqrt {25 + 56} }}{2}$
$\Rightarrow x = \dfrac{{5 \pm \sqrt {81} }}{2} \\ \Rightarrow x = \dfrac{{5 \pm 9}}{2} \;$
So we have
$\Rightarrow x = \dfrac{{5 + 9}}{2} = \dfrac{{14}}{2}$ and $x = \dfrac{{5 - 9}}{2} = \dfrac{{ - 4}}{2}$
On simplifying we get $x = 7$ and $x = - 2$
Hence the quadratic equation is correct.
Therefore the quadratic equation for the roots -2 and 7 is ${x^2} - 5x - 14$
So, the correct answer is “ $x = 7$ and $x = - 2$ ”.

Note: The equation is a quadratic equation. The roots can be solved by using the sum product rule. This defines as for the general quadratic equation $a{x^2} + bx + c$ , the product of $a{x^2}$ and c is equal to the sum of bx of the equation by using this we can form the equation and by using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ we can determine the roots for the equation.