First slide

Theory of expressions

Question

If the roots of the equation $x^{2} - 2 ax + a^{2} + a - 3 = 0$ are real and less than 3, then

Moderate

A

a < 2
B

$2 \leq a \leq 3$
C

$3 < a \leq 4$
D

a > 4

Solution

Given equation is $x^{2} - 2 ax + a^{2} + a - 3 = 0$ If roots are real then, $D \geq 0$
$\begin{array}{lrl} \Rightarrow 4 a^{2} - 4 (a^{2} + a - 3) \geq 0 \\ \Rightarrow - a + 3 \geq 0 \\ \Rightarrow a - 3 \leq 0 \Rightarrow a \leq 3 \end{array}$
As roots are less than 3, hence $f (3) > 0$
$\begin{array}{lr} 9 - 6 a + a^{2} + a - 3 > 0 \\ \Rightarrow a^{2} - 5 a + 6 > 0 \\ \Rightarrow (a - 2) (a - 3) > 0 \\ \Rightarrow Either a < 2 or a > 3 . \end{array}$
Hence, only a < 2 satisfy

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