In a quadratic equation $a x^{2} + b x + c = 0$ , the product of the roots of the equation is ____________.

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Solution

The quadratic equation given is $a x^{2} + b x + c = 0$
Roots of the above equation is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Let $x_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$ and $x_{2} = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}$
Product of $x_{1}$ and $x_{2}$ $= x_{1} \times x_{2}$
$= (\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}) (\frac{- b - \sqrt{b^{2} - 4 a c}}{2 a})$
$= \frac{b^{2} - (b^{2} - 4 a c)}{4 a^{2}}$
$= \frac{c}{a}$
$∴$ product of roots $= \frac{c}{a}$

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