x(2x-5)=x+6

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

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

Reorder the terms:
x(-5 + 2x) = x + 6
(-5 * x + 2x * x) = x + 6
(-5x + 2x2) = x + 6

Reorder the terms:
-5x + 2x2 = 6 + x

Solving
-5x + 2x2 = 6 + x

Solving for variable 'x'.

Reorder the terms:
-6 + -5x + -1x + 2x2 = 6 + x + -6 + -1x

Combine like terms: -5x + -1x = -6x
-6 + -6x + 2x2 = 6 + x + -6 + -1x

Reorder the terms:
-6 + -6x + 2x2 = 6 + -6 + x + -1x

Combine like terms: 6 + -6 = 0
-6 + -6x + 2x2 = 0 + x + -1x
-6 + -6x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
-6 + -6x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + -3x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
-3 + -3x + x2 = 0

Solving
-3 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 3 + 2.25

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

Combine like terms: 3 + 2.25 = 5.25
2.25 + -3x + x2 = 5.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
x + -1.5 = 2.291287847

Simplifying
x + -1.5 = 2.291287847

Reorder the terms:
-1.5 + x = 2.291287847

Solving
-1.5 + x = 2.291287847

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 = 2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 2.291287847 + 1.5
x = 2.291287847 + 1.5

Combine like terms: 2.291287847 + 1.5 = 3.791287847
x = 3.791287847

Simplifying
x = 3.791287847

Subproblem 2
x + -1.5 = -2.291287847

Simplifying
x + -1.5 = -2.291287847

Reorder the terms:
-1.5 + x = -2.291287847

Solving
-1.5 + x = -2.291287847

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 = -2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -2.291287847 + 1.5
x = -2.291287847 + 1.5

Combine like terms: -2.291287847 + 1.5 = -0.791287847
x = -0.791287847

Simplifying
x = -0.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.791287847, -0.791287847}
Solution
x = {3.791287847, -0.791287847}