First slide

Theory of expressions

Question

The set of possible values of $a$ for which $x^{2} - (a^{2} - 5 a + 5) x + (2 a^{2} - 3 a - 4) = 0$ has roots whose sum and product are both less than 1, is

Moderate

A

(-1, 5/2)
B

(1, 4)
C

[1, 5/2]
D

(1, 5/2)

Solution

We have,
$\begin{array}{l} a^{2} - 5 a + 5 < 1 and 2 a^{2} - 3 a - 4 < 1 \\ \Rightarrow a^{2} - 5 a + 4 < 0 and 2 a^{2} - 3 a - 5 < 0 \\ \Rightarrow (a - 1) (a - 4) < 0 and (2 a - 5) (a + 1) < 0 \end{array} \Rightarrow 1 < a < 4 and - 1 < a < \frac{5}{2} \Rightarrow 1 < a < \frac{5}{2}$

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