First slide

Theory of equations

Question

If $α, β$ are the roots of ${ax}^{2} + bx + c = 0$ and $α + h, β + h$ are the roots of ${px}^{2} + qx + r = 0$ then h =

Moderate

A

$- \frac{1}{2} (\frac{a}{b} - \frac{p}{q})$
B

$(\frac{b}{a} - \frac{q}{p})$
C

$\frac{1}{2} (\frac{b}{a} - \frac{q}{p})$
D

none of these

Solution

We have
$\begin{aligned} α + β = - \frac{b}{a}, αβ = \frac{c}{a} \\ α + h + β + h = - \frac{q}{p} \\ \Rightarrow α + β + 2 h = - \frac{q}{p} \end{aligned}$
$\begin{aligned} \Rightarrow - \frac{b}{a} + 2 h = \frac{- q}{p} \\ \Rightarrow h = \frac{1}{2} (\frac{b}{a} - \frac{q}{p}) \end{aligned}$

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