# 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