Hereย you will learn what is the relation between roots and coefficients of quadratic equation with examples.
Letโs begin โ
The general form of quadratic equation is \(ax^2 + bx + c\) = 0,ย a \(\ne\) 0.
The root of the given equation can be found by using the formula :
x = \(-b \pm \sqrt{b^2 โ 4ac}\over 2a\)
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) \(|\alpha โ \beta|\) = \(\sqrt{D}\over | a |\)
where D = \(b^2 โ 4ac\)
(b) A quadratic equation whose roots are \(\alpha\) and \(\beta\) is \((x โ \alpha)\) \((x โ \beta)\) = 0 i.e.
\(x^2 โ (\alpha + \beta)x + \alpha\beta\) = 0
i.e. \(x^2\) โ (sum of roots) x + product of roots = 0.
Example : If \(\alpha\) and \(\beta\) are the roots of a quadratic equation \(x^2 โ 3x + 5\) = 0. Find the sum of roots and product of roots.
Solution : We have, \(x^2 โ 3x + 5\) = 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,
\(x^2\) โ (sum of roots) x + product of roots = 0.
\(x^2 โ (5)x + (6)\) =0 \(\implies\) \(x^2 โ 5x + 6\) = 0
