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Nature of Roots of a Cubic Polynomial Using Derivatives

9 x2+4=0

Question

9x² + 4 = 0

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Solution

Given: $9 x^{2} + 4 = 0$

$9 x^{2} + 4 = 0 \Rightarrow (3 x)^{2} + 2^{2} = 0 \Rightarrow (3 x)^{2} - (2 i)^{2} = 0 \Rightarrow (3 x + 2 i) (3 x - 2 i) = 0 [(a^{2} - b^{2}) = (a + b) (a - b)]$
$\Rightarrow (3 x + 2 i) = 0$ or, $(3 x - 2 i) = 0$
$\Rightarrow 3 x = - 2 i$ or $3 x = 2 i$
$\Rightarrow x = - \frac{2 i}{3}$ or $x = \frac{2 i}{3}$
Hence, the roots of the equation are $\frac{2 i}{3} and - \frac{2 i}{3} .$

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