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Solving a Quadratic Equation by Completion of Squares Method

If α, β are...

Question

If $α, β$ are zeroes of the quadratic polynomial $a x^{2} + b x + c$ then find the value of $α^{2} - β^{2}$ .

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Solution

$a x^{2} + b x + c = 0$
$α + β = - \frac{b}{a} α β = \frac{c}{a}$
${(α + β)}^{2} = \frac{b^{2}}{a^{2}}$
${(α - β)}^{2} = {(α + β)}^{2} - 4 α β$
$= \frac{b^{2}}{a^{2}} - \frac{4 c}{a} = \frac{b^{2} - 4 a c}{a^{2}}$
$α - β = \frac{\sqrt{b^{2} - 4 a c}}{a}$
$α^{2} - β^{2} = (α + β) (α - β) = \frac{- b}{a} (\frac{\sqrt{b^{2} - 4 a c}}{a})$
$= \frac{- b \sqrt{b^{2} - 4 a c}}{a^{2}}$

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