First slide

Theory of expressions

Question

$Let α, β are roots of x^{2} - b x = b (b > 0) and | α |, | β | are roots of x^{2} + p x + q = 0 . The minimum value of (p^{2} - 8 q)$ is equal to

Moderate

A

-6
B

-4
C

0
D

4

Solution

$α and β are roots of x^{2} - b x = b$
$\begin{array}{l} ∴ α + β = b, α β = - b \\ | α | and | β | are roots of x^{2} + p x + q = 0 \\ \begin{array}{ll} ∴ & | α | + | β | = - p, | α | | β | = q \\ ∴ & p^{2} - 8 q = | α |^{2} + | β |^{2} - 6 | α | | β | \\ = (α + β)^{2} - 2 α β - 6 | α | | β | \\ = b^{2} + 2 b + 6 (- b) = (b^{2} - 4 b) = (b - 2)^{2} - 4 \end{array} \end{array}$
$So, minimum value of (p^{2} - 8 q) is - 4 .$

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