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 $-\dfrac{7}{3}$.

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