Solve the following equation that has equal roots:
$8 x^{4} + 4 x^{3} - 18 x^{2} + 11 x - 2 = 0$ .

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Solution

Given equation is : $8 x^{4} + 4 x^{3} - 18 x^{2} + 11 x - 2 = 0$

Consider $f (x) = 8 x^{4} + 4 x^{3} - 18 x^{2} + 11 x - 2$

$∴ f^{'} (x) = 32 x^{3} + 12 x^{2} - 36 x + 11$

Now, HCF of $f (x)$ and $f^{'} (x)$ is $(x - \frac{1}{2})$ . Hence $\frac{1}{2}$ is a double root of $f (x) = 0$

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

$∴$ roots of the given equation are $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, - 2$

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