Solve : $3 x^{3} + x^{2} - 2 = 0$ and $2 x^{2} + 5 x + 5 = 0$

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Solution

Given,

$3 x^{3} + x^{2} - 2 = 0$

Let, $f (x) = 3 x^{3} + x^{2} - 2$

$f^{'} (x) = 9 x^{2} + 2 x$

substituing differentiation to zero, we get,

$9 x^{2} + 2 x = 0$

$\Rightarrow 9 x^{2} = - 2 x$

$∴ x = - \frac{2}{9}$

Given,

$2 x^{2} + 5 x + 5 = 0$

$x_{1, 2} = \frac{- 5 \pm \sqrt{5^{2} - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

solving the above equation, we get,

$x_{1, 2} = - \frac{5}{4} \pm i \frac{\sqrt{15}}{4}$

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