Solve for x over the real numbers = complete the square
x + 5 = 0.5 (x + 3)^2
0.5 (x + 3)^2 = 1/2 (x + 3)^2:
x + 5 = 1/2 (x + 3)^2
Expand out terms of the right hand side:
x + 5 = x^2/2 + 3 x + 9/2
Subtract x^2/2 + 3 x + 9/2 from both sides:
-x^2/2 - 2 x + 1/2 = 0
Multiply both sides by -2:
x^2 + 4 x - 1 = 0
Add 1 to both sides:
x^2 + 4 x = 1
Add 4 to both sides:
x^2 + 4 x + 4 = 5
Write the left hand side as a square:
(x + 2)^2 = 5
Take the square root of both sides:
x + 2 = sqrt(5) or x + 2 = -sqrt(5)
Subtract 2 from both sides:
x = sqrt(5) - 2 or x + 2 = -sqrt(5)
Subtract 2 from both sides:
Answer: x = sqrt(5) - 2 or x = -2 - sqrt(5)
