Math Doubts

Product of Roots of a Quadratic equation

The product of the multiplication of the two roots of a quadratic equation is called the product of the roots of a quadratic equation.

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 $\dfrac{c}{a}$.


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

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
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.

Learn more