First slide

Theory of expressions

Question

$The values of p for which one root of the equation x^{2} - (p + 1) x + p^{2} + p = 8 exceeds 2 and the other is lesser than 2 are given by$

Moderate

A

$3 < p < 10$
B

$p \leq - 2$
C

$p \geq 10$
D

$- 2 < p < 3$

Solution

$Let f (x) = x^{2} - (p + 1) x + p^{2} + p - 8$
Since one root is less than 2 and other is more than 2, 2lies between the roots
$\begin{array}{ll} \Rightarrow f (2) < 0 \\ \Rightarrow 4 - 2 (p + 1) + p^{2} + p - 8 < 0 \\ \Rightarrow p^{2} - p - 6 < 0 \\ \Rightarrow (p - 3) (p + 2) < 0 \\ \Rightarrow p \in (- 2, 3) \end{array}$

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