# Product of Roots of a Quadratic equation

The product of the roots (or zeros) of a quadratic equation is called the product of the roots of a quadratic equation.

## Introduction

According to quadratic formula, the roots of quadratic equation $ax^2+bx+c = 0$ are represented by Alpha ($\alpha$) and Beta ($\beta$).

$\alpha$ $\,=\,$ $\dfrac{-b + \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

$\beta$ $\,=\,$ $\dfrac{-b \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

Multiply the two roots of standard form quadratic equation to obtain the product of them

$\alpha \times \beta$ $=$ $\Bigg(\dfrac{-b + \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}\Bigg)$ $\times$ $\Bigg(\dfrac{-b \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}\Bigg)$

The product of roots $\alpha$ and $\beta$ is written as $\alpha \beta$ in simple form.

$\implies$ $\alpha \beta$ $=$ $\Bigg(\dfrac{(-b) + \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}\Bigg)$ $\times$ $\Bigg(\dfrac{(-b) \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}\Bigg)$

The two multiplying factors are fractions. So, multiply them by using the method of multiplying fractions.

$\implies \alpha \beta = \dfrac{\Big((-b) + \sqrt{b^{\displaystyle 2} \,-\, 4ac} \Big) \times \Big((-b) \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}\Big)}{2a \times 2a}$

The two factors in the numerators have same terms with opposite signs.

$\implies$ $\alpha \beta = \dfrac{ {\Big(-b\Big)}^2 \,-\, {\Big(\sqrt{b^{\displaystyle 2} \,-\, 4ac}\Big)}^2 }{4a^2}$

$\implies$ $\alpha \beta = \dfrac{ b^2 \,-\, \Big(b^2 \,-\, 4ac \Big) }{4a^2}$

$\implies$ $\alpha \beta = \dfrac{ b^2 \,-\, b^2 + 4ac }{4a^2}$

$\implies$ $\require{cancel} \alpha \beta = \dfrac{\cancel{b^2} \,-\, \cancel{b^2} + 4ac }{4a^2}$

$\implies$ $\alpha \beta = \dfrac{4ac}{4a^2}$

$\implies$ $\require{cancel} \alpha \beta = \dfrac{\cancel{4a}c}{\cancel{4a}^2}$

$\,\,\, \therefore \,\,\,\,\,\, \alpha \beta = \dfrac{c}{a}$

Therefore, the product of two roots of the quadratic equation $ax^2+bx+c = 0$ is $\dfrac{c}{a}$.

### Example

Find the product of roots of the quadratic equation $3x^2 \,-\, 4x \,-\, 7 = 0$

Compare the given quadratic equation with general form quadratic equation $ax^2 + bx + c = 0$.

$a = 3$, $b = -4$ and $c = -7$.

The product of roots of quadratic equation is $\dfrac{c}{a}$

$\dfrac{c}{a} = \dfrac{-7}{3}$

Therefore, the product of roots of the quadratic equation $3x^2 \,-\, 4x \,-\, 7 = 0$ is $-\dfrac{7}{3}$.

Email subscription
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
###### Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.