Let
f(x) = 2x + 7 if x < -2
f(x) = -x^2 - x + 2 if x >= -2
Find the sum of all values of x such that f(x) = -5.
+130477 f(x) = 2x + 7 if x < - 2 f(x) = -x^2 - x + 2 if x > = -2
-5 = 2x + 7 -5 = -x^2 - x + 2 rewrite as
-5 - 7 = 2x x^2 + x - 5 - 2 = 0
-12 = 2x x^2 + x - 7 = 0 complete the square on x
-12/2 = x = -6 x^2 + x + 1/4 = 7 + 1/4
(x + 1/2) = 29/4 take the positive root
x + 1/2 = [ sqrt (29) ] / 2
x = [ sqrt (29) / 2 - 1/2
The sum of the x values that make this true are -6 + [ sqrt (29) ] /2 - 1/2 = -13/2 + [ sqrt (29)] / 2 ≈ -3.807
