First slide

Theory of equations

Question

$Let α and β be the roots of the quadratic equation p x^{2} + q x + r = 0, where α β = 99 . If p, q and r (taken in that order)$
$are in arithmetic progression, then (α + β) is equal to$

Moderate

Solution

$\begin{array}{l} α + β = \frac{- q}{p}; α β = \frac{r}{p} \\ Also, 2 q = p + r \\ \Rightarrow \frac{2 q}{p} = 1 + \frac{r}{p} \\ \Rightarrow - 2 (α + β) = 1 + α β = 1 + 99 = 100 \\ \Rightarrow (α + β) = - 50 \end{array}$

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