Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$x^{2} + 6 x - (a^{2} + 2 a - 8) = 0$

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Solution

$x^{2} + 6 x - (a^{2} + 2 a - 8) = 0$
Solving the Q.E in x,we have

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

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

$D = 6^{2} - 4 (- (a 2 + 2 a - 8))$

$= 36 + 4 a^{2} + 8 a - 32 = 36 + 4 a^{2} + 8 a - 32 = 4 a^{2} + 8 a + 4 = 4 (a + 1)^{2} > 0, so solution exists$

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

$x = \frac{- 6 \pm \sqrt{4 (a + 1)^{2}}}{2}$

$x = \frac{- 6 \pm 2 (a + 1)}{2}$

$x = - 3 + a + 1 o r x = - 3 - a - 1$

$x = a - 2 o r x = - a - 4$

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