First slide

Theory of equations

Question

$The greatest value of a non-negative real number λ for which both the equations 2 x^{2} + (λ - 1) x + 8 = 0 and x^{2} - 8 x + λ + 4 = 0$
$have real roots is$

Moderate

A

9
B

12
C

15
D

16

Solution

$For real roots, (λ - 1)^{2} - 64 \geq 0 and 64 - 4 (λ + 4) \geq 0$
$\begin{array}{l} \Rightarrow (λ - 1)^{2} \geq 64 and 48 - 4 λ \geq 0 \\ \Rightarrow λ - 1 \geq 8 or λ - 1 \leq - 8 and 12 \geq λ \\ \Rightarrow λ \geq 9 or λ \leq - 7 and λ \leq 12 \\ Hence greatest value of λ = 12 . \end{array}$

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