Find the values of $k$ for which the given equation has real roots
$9 x^{2} - 3 k x + 4 = 0$

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Solution

We have $9 x^{2} - 3 k x + 4 = 0$
On comparing the above quadratic equation with standard one $a x^{2} + b x + c = 0$ , we have

Here, $a = 9, b = - 3 k, c = 4$

$∴ D = b^{2} - 4 a c = {9 k}^{2} - 4 \times 9 \times 4 = 9 k^{2} - 144$

The given equation will have real roots, if
$D \geq 0$

$\Rightarrow 9 k^{2} - 144 \geq 0$
$\Rightarrow k \leq - 4 o r k \geq 4 [∵ x^{2} - a^{2} \geq 0 \Rightarrow x \leq - a o r x \geq a]$

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