If equation $(a - 1) x^{2} + (a^{2} + a - 2) x + (a^{2} - 3 a + 2) = 0$ has more than two roots then $a$ is

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Solution

$(a - 1) x^{2} + (a^{2} + a - 2) x + (a^{2} - 3 a + 2) = 0$
$\Rightarrow (a - 1) x^{2} + (a^{2} + 2 a - a - 2) x + (a^{2} - 2 a - a + 2) = 0$
$\Rightarrow (a - 1) x^{2} + (a + 2) (a - 1) x + (a - 2) (a - 1) = 0$ as it is an identity.
$\Rightarrow a - 1 = 0 \Rightarrow a = 1$
$\Rightarrow (a + 2) (a - 1) = 0 \Rightarrow a = 1, - 2$
$\Rightarrow (a - 2) (a - 1) = 0 \Rightarrow a = 1, 2$
So, ${1} \cap {1, - 2} \cap {1, 2} = 1$
So $a = 1$

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