Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$x^{2} + 5 x - (a^{2} + a - 6) = 0$

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Solution

$x^{2} + 5 x - (a^{2} + a - 6) = 0$
Solving the Q.E in x,we have

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2}$

Discriminant, $D = b^{2} - 4 a c$

$D = 5^{2} - 4 (- (a^{2} + a - 6))$

$= 25 + 4 a^{2} + 4 a - 24 = 4 a 2 + 4 a + 1 = (2 a + 1)^{2} > 0, so solution exists$

$x = \frac{- b \pm \sqrt{D}}{2}$

$x = \frac{- 5 \pm \sqrt{(2 a + 1)^{2}}}{2}$

$x = \frac{- 5 \pm 2 a + 1}{2}$

$x = \frac{- 4 \pm 2 a}{2}$

$x = - 2 \pm a$

$x = - 2 + a o r x = - 2 - a$

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