First slide

Theory of equations

Question

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

Moderate

A

$\frac{1}{α}, \frac{1}{β}$
B

$\frac{1}{α}, β$
C

$\frac{α}{1 + α}, \frac{β}{1 + β}$
D

$α, \frac{1}{β}$

Solution

We can write (1) as
$\begin{array}{l} a {(- \frac{x}{x - 1})}^{2} + b (\frac{- x}{x - 1}) + c = 0 \\ \Rightarrow \frac{- x}{x - 1} = α, β \Rightarrow x = \frac{α}{1 + α}, \frac{β}{1 + β} \end{array}$

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