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Solving a Quadratic Equation by Completion of Squares Method

If α and ...

Question

If $α$ and $β$ are the roots of the equation $x^{2} - 6 x + 8 = 0,$ then find the values of
(i) $α^{2} + β^{2}$ (ii) $1 / α + 1 / β$ (iii) $α - β (α > β)$

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Solution

From the given equation we get $α + β = 6$ and $α β = 8$
(i) $α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = {(6)}^{2} - 2 (8) = 20$
(ii) $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = \frac{6}{8} = \frac{3}{4}$
(iii) ${(α - β)}^{2} = {(α + β)}^{2} - 4 α β$
$\Rightarrow (α - β) = \pm \sqrt{{(α + β)}^{2} - 4 α β} = \sqrt{6^{2} - 4 (8)}$
$\Rightarrow (α - β) = \pm 2$
$∴ α - β = 2 (∵ α > β)$

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