Question

Find the quadratic equation whose roots are $-5$ and $-2$ .

Answer 1

Hint: First assume a quadratic equation in any form. Apply conditions for sum of roots and product of roots. By this find the values of variables in the quadratic equation. These values can be substituted back into the equation. This equation is the required result.

Complete step-by-step answer:
Given the roots of the quadratic equation are written in the form of $-5$ and $-2$ .
Let the two roots be denoted by variables m, n
Let us assume the first root, $-5$ is denoted by m.
Let us assume the second root, $-2$ is denoted by n.
Let us assume the required quadratic equation as:
$a{{x}^{2}}+bx+c=0$ ……………………………….. (1)
By dividing with ‘a’ on both sides, we get it as:
$\dfrac{a{{x}^{2}}+bx+c}{a}=\dfrac{0}{a}$
By basic knowledge of the fractions, we know the relation:
$\dfrac{A+B+C}{D}=\dfrac{A}{D}+\dfrac{B}{D}+\dfrac{C}{D}$
By using this here, we can write the equation as:
$\dfrac{a{{x}^{2}}}{a}+\dfrac{bx}{a}+\dfrac{c}{a}=\dfrac{0}{a}$
By simplifying the above equation, we can write it as:
$\dfrac{a{{x}^{2}}}{a}+\dfrac{bx}{a}+\dfrac{c}{a}=0$
By cancelling the common terms, we can write it as:
${{x}^{2}}+\dfrac{b}{a}x+\dfrac{c}{a}=\dfrac{0}{a}$ ……………………. (2)
By basic knowledge of equation, we know that for a quadratic equation, we have relations such as:
Sum of roots of equation (1), is equal to $\dfrac{-b}{a}$ .
Product of roots of equation (1), is equal to $\dfrac{c}{a}$ .
Writing the sum of roots statements mathematically, we get it as:
$m+n=\dfrac{-b}{a}$
By substituting m, n we get the above equation in form of:
$-5-2=\dfrac{-b}{a}$
By cancelling the minus sign we can write the equation:
$\dfrac{b}{a}=7$ ……………………….. (3)
Writing product of roots statement mathematically, we get it as:
$m.n=\dfrac{c}{a}$
By substituting the values of m, n, we get it as:
$\left( -5 \right).\left( -2 \right)=\dfrac{c}{a}$
By simplifying equation (4) (3) in equation (2) we get:
${{x}^{2}}+7x+10=0$ is an equation with roots $-5,-2$ .

Note: Be careful while taking the sum of roots formula itself as ‘-‘ sign. If you miss that then you will get $\dfrac{b}{a}$ value as -7. Then the whole equation you present will be wrong. While taking product students confuse and write $-5\times -2$ is $-10$ but it is +10. So, be careful at this step.