4x+24=4x^2-8-32

Solution for 4x+24=4x^2-8-32 equation:

Simplifying
4x + 24 = 4x2 + -8 + -32

Reorder the terms:
24 + 4x = 4x2 + -8 + -32

Reorder the terms:
24 + 4x = -8 + -32 + 4x2

Combine like terms: -8 + -32 = -40
24 + 4x = -40 + 4x2

Solving
24 + 4x = -40 + 4x2

Solving for variable 'x'.

Reorder the terms:
24 + 40 + 4x + -4x2 = -40 + 4x2 + 40 + -4x2

Combine like terms: 24 + 40 = 64
64 + 4x + -4x2 = -40 + 4x2 + 40 + -4x2

Reorder the terms:
64 + 4x + -4x2 = -40 + 40 + 4x2 + -4x2

Combine like terms: -40 + 40 = 0
64 + 4x + -4x2 = 0 + 4x2 + -4x2
64 + 4x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
64 + 4x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(16 + x + -1x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(16 + x + -1x2)' equal to zero and attempt to solve:

Simplifying
16 + x + -1x2 = 0

Solving
16 + x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-16 + -1x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -1x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -1x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -1x + x2 = 0 + 16
-1x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-1x + x2 = 16

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 16 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 16 + 0.25

Combine like terms: 16 + 0.25 = 16.25
0.25 + -1x + x2 = 16.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.031128874 and -4.031128874.

Subproblem 1
x + 0.5 = 4.031128874

Simplifying
x + 0.5 = 4.031128874

Reorder the terms:
0.5 + x = 4.031128874

Solving
0.5 + x = 4.031128874

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.031128874 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.031128874 + -0.5
x = 4.031128874 + -0.5

Combine like terms: 4.031128874 + -0.5 = 3.531128874
x = 3.531128874

Simplifying
x = 3.531128874

Subproblem 2
x + 0.5 = -4.031128874

Simplifying
x + 0.5 = -4.031128874

Reorder the terms:
0.5 + x = -4.031128874

Solving
0.5 + x = -4.031128874

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.031128874 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.031128874 + -0.5
x = -4.031128874 + -0.5

Combine like terms: -4.031128874 + -0.5 = -4.531128874
x = -4.531128874

Simplifying
x = -4.531128874

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.531128874, -4.531128874}
Solution
x = {3.531128874, -4.531128874}