Product of Roots of a Quadratic equation

$\alpha$ and $\beta$ are two roots of the quadratic equation $ax^{\displaystyle 2} + bx + c = 0$ and they can be expressed in terms of literal coefficients of the same quadratic equation.

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

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

Multiply two roots of the 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 and write $–b$ inside the brackets in numerator of both multiplying factors.

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

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 multiplying factors of numerators consists of same terms but they contain opposite signs. The product of two binomials having opposite signs can be used here.

In other words, $(x+y)(x-y) = x^2 \,-\, y^2$. If $x = -b$ and $y = \sqrt{b^{\displaystyle 2} \,-\, 4ac}$, the product can be written in same form.

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

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

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

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

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

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

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

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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