Relation Between Roots and Coefficients of Quadratic Equation

Here you will learn what is the relation between roots and coefficients of quadratic equation with examples.

Let’s begin –

The general form of quadratic equation is $ax^2 + bx + c$ = 0,  a $\ne$ 0.

The root of the given equation can be found by using the formula :

x = $-b \pm \sqrt{b^2 – 4ac}\over 2a$

Relation Between Roots and Coefficients of Quadratic Equation

(a) Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c$ = 0, then

(i) Sum of roots is $\alpha$ + $\beta$ = $-b\over a$

(ii) Product of roots is $\alpha$ $\beta$ = $c\over a$

(iii) $|\alpha – \beta|$ = $\sqrt{D}\over | a |$

where D = $b^2 – 4ac$

(b) A quadratic equation whose roots are $\alpha$ and $\beta$ is $(x – \alpha)$ $(x – \beta)$ = 0 i.e.

$x^2 – (\alpha + \beta)x + \alpha\beta$ = 0

i.e. $x^2$ – (sum of roots) x + product of roots = 0.

Example : If $\alpha$ and $\beta$ are the roots of a quadratic equation $x^2 – 3x + 5$ = 0. Find the sum of roots and product of roots.

Solution : We have, $x^2 – 3x + 5$ = 0

Sum of Roots = $\alpha$ + $\beta$ = $-b\over a$ = 3

Product of Roots = $\alpha$$\beta$ = $c\over a$ = 5

Example : Find the quadratic equation whose sum of roots is 5 and product of roots is 6.

Solution : By using the formula,

$x^2$ – (sum of roots) x + product of roots = 0.

$x^2 – (5)x + (6)$ =0 $\implies$ $x^2 – 5x + 6$ = 0