Answer: We can find (f+g)(x) by adding f(x) and g(x):
(f+g)(x) = f(x) + g(x) = (8-x²) + (x-3) = -x² + x + 5
To find the roots of (f+g)(x), we need to set it equal to zero and solve for x:
-x² + x + 5 = 0
We can use the quadratic formula to solve for x:
x = (-b ± sqrt(b² - 4ac)) / 2a
Where a = -1, b = 1, and c = 5:
x = (-1 ± sqrt(1² - 4(-1)(5))) / 2(-1)
x = (-1 ± sqrt(1 + 20)) / -2
x = (-1 ± sqrt(21)) / -2
So the roots of (f+g)(x) are:
x = (-1 + sqrt(21)) / -2
x = (-1 - sqrt(21)) / -2
These roots cannot be simplified any further, so we leave the answer in this form.
Explanation: