Final answer:
The quadratic equation -x^2 + x + 13 = -16 is first transformed into standard form, x^2 - x - 3 = 0, and then solved using the quadratic formula to find the roots x = (1 + √13) / 2 and x = (1 - √13) / 2.
Step-by-step explanation:
To solve the quadratic equation -x^2 + x + 13 = -16, we must first rearrange it into the standard form of a quadratic equation, which is ax^2 + bx + c = 0. To do this, we add 16 to both sides of the equation, resulting in x^2 - x - 3 = 0. We now apply the quadratic formula, which is x = [-b ± √(b^2 - 4ac)] / (2a), to find the roots of the equation.
By substituting a = 1, b = -1, and c = -3 into the formula, we find:
x = [-(-1) ± √((-1)^2 - 4(1)(-3))] / (2*1)
x = [1 ± √(1 + 12)] / 2
x = [1 ± √(13)] / 2
This results in two potential solutions for x:
- x = (1 + √13) / 2
- x = (1 - √13) / 2
These are the roots of the quadratic equation.