Solve the equation: $- {(x - 2)}^{2} - 2 = 7$ and find the sum of its roots.

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Solution

Solve $- (x - 2)^{2} - 2 = 7$ as follows:

$- (x^{2} + 4 - 4 x) - 2 = 7$
$- x^{2} - 4 + 4 x - 2 = 7$
$- x^{2} + 4 x = 13$
$- x^{2} + 4 x - 13 = 0$
Now we find the roots of above quadratic equation as shown below:
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$= \frac{- (- 4) \pm \sqrt{{(4)}^{2} - 4 (1) (13)}}{2 (1)}$
$= \frac{4 \pm \sqrt{16 - 52}}{2}$
$= \frac{4 \pm \sqrt{- 36}}{2}$
$= \frac{4 \pm 6 i}{2}$
$= 2 \pm 3 i$
Therefore, $x = 2 \pm 3 i$ .
Thus the sum of the roots of the equation is : $2 + 3 i$ + $2 - 3 i$
$= 4$

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