Solve the following equations which have equal roots:
$x^{4} - 9 x^{2} + 4 x + 12 = 0$ .

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Solution

Give equation, $x^{4} - 9 x^{2} + 4 x + 12 = 0$

Consider $f (x) = x^{4} - 9 x^{2} + 4 x + 12$

$∴ f^{'} (x) = 4 x^{3} - 18 x + 4$

Now, HCF of $f (x)$ and $f^{'} (x)$ is $(x - 2)$ . Hence 2 is a double root of $f (x) = 0$

$f (x)$ can be factored as $(x - 2)^{2} (x^{2} + 4 x + 3) or (x - 2)^{2} (x + 3) (x + 1)$

$∴$ roots of the given equation are $- 3, - 1, 2, 2$

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