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 ax2+bx+c = 0, a ≠ 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