First slide

Theory of equations

Question

lf $α, β$ are the roots of $x^{2} + b x - c - 0$ , the the equation whose roots are $b$ and $c,$ is

Moderate

A

$x^{2} + α x - β = 0$
B

$x^{2} - x (α + β + α β) - α β (α + β) = 0$
C

$x^{2} + (α + β - α β) x - α β (α + β) = 0$
D

$x^{2} + x (α + β + α β) + α β (α + β) = 0$

Solution

Since $α, β$ are roots of $x^{2} + b x - c = 0$
$∴ α + β = - b, α β = - c$
The equation whose roots are $b, c$ is
$\begin{array}{l} x^{2} - x (b + c) + b c = 0 \\ \Rightarrow x^{2} - x (- α - β - α β) + α β (α + β) = 0 \\ \Rightarrow x^{2} + x (α + β + α β) + α β (α + β) = 0 \end{array}$

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