First slide

Theory of equations

Question

If $α, β$ are the roots of $x^{2} + px + q = 0$ and $γ, δ$ are the roots of $x^{2} + px + r = 0$ then $\frac{(α - γ) (α - δ)}{(β - γ) (β - δ)} =$

Moderate

A

1
B

q
C

r
D

q + r

Solution

$α, β$ be the roots of $x^{2} + px + q = 0$
$γ, δ$ be the roots of $x^{2} + px + r = 0$
$\begin{array}{l} α + β = - p \\ αβ = q \\ γ + δ = - p \\ γδ = r \end{array}$
Now, $(α - γ) (α - δ) = α^{2} - α (γ + δ) + γδ$
$\begin{array}{l} = α^{2} - α (α + β) + r \\ = - αβ + r = - q + r \end{array}$
By symmetry $(β - γ) (β - δ) = - q + r$
Hence, $\frac{(α - γ) (α - δ)}{(β - γ) (β - δ)} = 1$

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