For the quadratic equation ${a x}^{2} + b x + c = 0$ drive the formula to find $x$ . Using the formula find the roots of $2 x^{2} + x - 528 = 0$ .

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Solution

$a x^{2} + b x + c = 0$
$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$
making this into a perfect square
$x^{2} + 2 \frac{b x}{2 a} + \frac{b^{2}}{(2 a)^{2}} + \frac{c}{a} = \frac{b^{2}}{(2 a)^{2}}$
$(x + \frac{b}{2 a})^{2} = \frac{b^{2} - 4 a c}{4 a^{2}}$
$x + \frac{b}{2 a} = \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}$
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$∴$ Hence derived.
$2 x^{2} + x - 528 = 0$
$x = \frac{- 1 \pm \sqrt{1 + 4 (2) (528)}}{4} = 8, \frac{- 33}{2}$

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