Question

select the correct answer.
what are the solutions of this quadratic equation?
x² - 8x = -65

○ x = 8 + 13i and x = 8 - 13i
○ x = 4 + 7i and x = 4 - 7i
○ x = -4 + 7i and x = -4 - 7i
○ x = -8 + 13i and x = -8 - 13i

Explanation:

Step1: Rewrite in standard form

Rearrange the equation to $ax^2+bx+c=0$:
$x^2 - 8x + 65 = 0$

Step2: Use quadratic formula

Recall formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, here $a=1$, $b=-8$, $c=65$:
$x=\frac{-(-8)\pm\sqrt{(-8)^2-4(1)(65)}}{2(1)}$

Step3: Calculate discriminant

Compute the value under the square root:
$\sqrt{64 - 260}=\sqrt{-196}=14i$

Step4: Simplify the expression

Substitute discriminant back and simplify:
$x=\frac{8\pm14i}{2}=4\pm7i$

Answer:

$x = 4 + 7\text{i and } x = 4 - 7\text{i}$