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

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