(2x-3)(3x+4)=5x-6

Solution for (2x-3)(3x+4)=5x-6 equation:

Simplifying
(2x + -3)(3x + 4) = 5x + -6

Reorder the terms:
(-3 + 2x)(3x + 4) = 5x + -6

Reorder the terms:
(-3 + 2x)(4 + 3x) = 5x + -6

Multiply (-3 + 2x) * (4 + 3x)
(-3(4 + 3x) + 2x * (4 + 3x)) = 5x + -6
((4 * -3 + 3x * -3) + 2x * (4 + 3x)) = 5x + -6
((-12 + -9x) + 2x * (4 + 3x)) = 5x + -6
(-12 + -9x + (4 * 2x + 3x * 2x)) = 5x + -6
(-12 + -9x + (8x + 6x2)) = 5x + -6

Combine like terms: -9x + 8x = -1x
(-12 + -1x + 6x2) = 5x + -6

Reorder the terms:
-12 + -1x + 6x2 = -6 + 5x

Solving
-12 + -1x + 6x2 = -6 + 5x

Solving for variable 'x'.

Reorder the terms:
-12 + 6 + -1x + -5x + 6x2 = -6 + 5x + 6 + -5x

Combine like terms: -12 + 6 = -6
-6 + -1x + -5x + 6x2 = -6 + 5x + 6 + -5x

Combine like terms: -1x + -5x = -6x
-6 + -6x + 6x2 = -6 + 5x + 6 + -5x

Reorder the terms:
-6 + -6x + 6x2 = -6 + 6 + 5x + -5x

Combine like terms: -6 + 6 = 0
-6 + -6x + 6x2 = 0 + 5x + -5x
-6 + -6x + 6x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-6 + -6x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-1 + -1x + x2) = 0

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

Simplifying
-1 + -1x + x2 = 0

Solving
-1 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The x term is -1x.  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 = 1 + 0.25

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

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1x + x2 = 1.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x + -0.5 = 1.118033989

Simplifying
x + -0.5 = 1.118033989

Reorder the terms:
-0.5 + x = 1.118033989

Solving
-0.5 + x = 1.118033989

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 = 1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 1.118033989 + 0.5
x = 1.118033989 + 0.5

Combine like terms: 1.118033989 + 0.5 = 1.618033989
x = 1.618033989

Simplifying
x = 1.618033989

Subproblem 2
x + -0.5 = -1.118033989

Simplifying
x + -0.5 = -1.118033989

Reorder the terms:
-0.5 + x = -1.118033989

Solving
-0.5 + x = -1.118033989

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 = -1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -1.118033989 + 0.5
x = -1.118033989 + 0.5

Combine like terms: -1.118033989 + 0.5 = -0.618033989
x = -0.618033989

Simplifying
x = -0.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.618033989, -0.618033989}
Solution
x = {1.618033989, -0.618033989}