Final answer:
To solve the quadratic equation (4x+5)² = -5(5+4x), we can recognize that the left side is a perfect square and simplify the equation accordingly.
By taking the square root, rearranging the terms, and applying the quadratic formula, we find that x can equal -2.5 or -1.25.
Step-by-step explanation:
We could solve this equation with the quadratic formula, but it is far easier to solve for x by recognizing that the left side of the equation is a perfect square; that is,
(4x+5)² = -5(5+4x)
The fraction is a perfect square, as is the 4.0 on the right.
So we can take the square root of both sides:
(4x+5) = ±√(-5)(5+4x)
Now we rearrange and solve (be sure you can follow each step):
4x+5 = ±√(-5)(5+4x)
4x+5 = ±√(-25-20x)
Squaring both sides again, we get:
16x² + 40x + 25 = -25 - 20x
16x² + 60x + 50 = 0
Now we use the quadratic formula to solve for the two possible values of x:
x = (-60 ± √(60² - 4(16)(50))) / (2(16))
x = (-60 ± √(3600 - 3200)) / 32
x = (-60 ± √400) / 32
x = (-60 ± 20) / 32
x = -80/32
= -2.5
x = -40/32
= -1.25