First slide

Theory of equations

Question

The equation $(a^{2} - a - 2) x^{2} + (a^{2} - 4) x + (a^{2} - 3 a + 2) = 0$ will have more than two solutions if a equal to

Moderate

A

2
B

3
C

-2
D

Not possible

Solution

Given , $(a^{2} - a - 2) x^{2} + (a^{2} - 4) x + (a^{2} - 3 a + 2) = 0 - - - - - (i)$
Since, Eq.(i) have more than two solutions.
So, it is an Identity.
$\begin{array}{l} ∴ a^{2} - a - 2 = a^{2} - 4 = a^{2} - 3 a + 2 = 0 \\ [∵ if {ax}^{2} + bx + c = 0 is an identity, then a = b = c = 0] \end{array}$

$\begin{array}{l} a^{2} - a - 2 = 0 \Rightarrow a = 2, - 1 \\ a n d a^{2} - 4 = 0 \Rightarrow a = \pm 2, \\ a n d a^{2} - 3 a + 2 = 0 \Rightarrow a = 1, 2 \\ t h e r e f o r e c o m m o n v a l u e o f a = 2 \end{array}$

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