Answer

Verified

410.1k+ views

**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.

**:**

__Complete step-by-step answer__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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How do you graph the function fx 4x class 9 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What organs are located on the left side of your body class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell