16(x^2-1)=24(2x+1)

Solution for 16(x^2-1)=24(2x+1) equation:

Simplifying
16(x2 + -1) = 24(2x + 1)

Reorder the terms:
16(-1 + x2) = 24(2x + 1)
(-1 * 16 + x2 * 16) = 24(2x + 1)
(-16 + 16x2) = 24(2x + 1)

Reorder the terms:
-16 + 16x2 = 24(1 + 2x)
-16 + 16x2 = (1 * 24 + 2x * 24)
-16 + 16x2 = (24 + 48x)

Solving
-16 + 16x2 = 24 + 48x

Solving for variable 'x'.

Reorder the terms:
-16 + -24 + -48x + 16x2 = 24 + 48x + -24 + -48x

Combine like terms: -16 + -24 = -40
-40 + -48x + 16x2 = 24 + 48x + -24 + -48x

Reorder the terms:
-40 + -48x + 16x2 = 24 + -24 + 48x + -48x

Combine like terms: 24 + -24 = 0
-40 + -48x + 16x2 = 0 + 48x + -48x
-40 + -48x + 16x2 = 48x + -48x

Combine like terms: 48x + -48x = 0
-40 + -48x + 16x2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(-5 + -6x + 2x2) = 0

Ignore the factor 8.
Subproblem 1
Set the factor '(-5 + -6x + 2x2)' equal to zero and attempt to solve:

Simplifying
-5 + -6x + 2x2 = 0

Solving
-5 + -6x + 2x2 = 0

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

Divide each side by '2'.
-2.5 + -3x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -3x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -3x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -3x + x2 = 0 + 2.5
-3x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-3x + x2 = 2.5

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 2.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 2.5 + 2.25

Combine like terms: 2.5 + 2.25 = 4.75
2.25 + -3x + x2 = 4.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 4.75

Calculate the square root of the right side: 2.179449472

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.179449472 and -2.179449472.

Subproblem 1
x + -1.5 = 2.179449472

Simplifying
x + -1.5 = 2.179449472

Reorder the terms:
-1.5 + x = 2.179449472

Solving
-1.5 + x = 2.179449472

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 2.179449472 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 2.179449472 + 1.5
x = 2.179449472 + 1.5

Combine like terms: 2.179449472 + 1.5 = 3.679449472
x = 3.679449472

Simplifying
x = 3.679449472

Subproblem 2
x + -1.5 = -2.179449472

Simplifying
x + -1.5 = -2.179449472

Reorder the terms:
-1.5 + x = -2.179449472

Solving
-1.5 + x = -2.179449472

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -2.179449472 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -2.179449472 + 1.5
x = -2.179449472 + 1.5

Combine like terms: -2.179449472 + 1.5 = -0.679449472
x = -0.679449472

Simplifying
x = -0.679449472

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.679449472, -0.679449472}
Solution
x = {3.679449472, -0.679449472}