First slide

Theory of equations

Question

If the two equations $x^{2} - cx + d = 0$ and $x^{2} - ax + b = 0$ have common root and the second has equal roots, then $2 (b + d) =$

Moderate

A

$0$
B

$a + c$
C

$ac$
D

$- ac$

Solution

Let roots of $x^{2} - cx + d = 0$ be $α, β$ then roots of
$x^{2} - ax + b = 0 be α, α$
$∴ α + β = c, αβ = d, α + α = a, a^{2} = b$
Hence $2 (b + d) = 2 (α^{2} + αβ) = 2 α (α + β) = ac$

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