First slide

Theory of equations

Question

If $α$ and $β$ arc roots of the equation ${ax}^{2} + bx + c = 0,$ then the roots of the equation $a (2 x + 1)^{2} - b (2 x + 1) (3 - x) + c (3 - x)^{2} = 0$ are

Moderate

A

$\frac{2 α + 1}{α - 3}, \frac{2 β + 1}{β - 3}$
B

$\frac{3 α + 1}{α - 2}, \frac{3 β + 1}{β - 2}$
C

$\frac{2 α - 1}{α - 2}, \frac{2 β + 1}{β - 2}$
D

None of these

Solution

$\begin{aligned} a \frac{(2 x + 1)^{2}}{(x - 3)^{2}} + b \frac{(2 x + 1)}{(x - 3)} + c = 0 \\ \Rightarrow \frac{2 x + 1}{x - 3} = α or \frac{2 x + 1}{x - 3} = β \end{aligned}$
$\begin{array}{l} or 2 x + 1 = αx - 3 α \\ or x (α - 2) = 1 + 3 α \\ or x = \frac{1 + 3 α}{α - 2}, \frac{1 + 3 β}{β - 2} \end{array}$

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