Question

solve ( x^2 - 16x + 60 = -12 ) by completing the steps. first, subtract (\boldsymbol{60}) from each side of the equation. next, add (\boldsymbol{64}) to each side of the equation to complete the square. now, write ( x^2 - 16x + 64 = -8 ) as (\boldsymbol{sqrt{(x - 8)^2} = -8}). take the square root of both sides to get the solutions (\boldsymbol{square pm square sqrt{2}i}).

Explanation:

Step1: Rewrite equation after prior steps

We start with the completed square form:
$$(x-8)^2 = -8$$

Step2: Simplify the right-hand side

Rewrite $-8$ as $8 \times (-1)$:
$$(x-8)^2 = 8i^2$$
where $i^2=-1$

Step3: Take square roots of both sides

$$x-8 = \pm \sqrt{8i^2}$$
Simplify $\sqrt{8i^2} = \sqrt{4 \times 2}i = 2\sqrt{2}i$

Step4: Solve for x

Isolate $x$ by adding 8 to both sides:
$$x = 8 \pm 2\sqrt{2}i$$

Answer:

$8$ $\pm$ $2$ $\sqrt{2}i$