(5x-2)(x-1)=x(x+1)

Solution for (5x-2)(x-1)=x(x+1) equation:

Simplifying
(5x + -2)(x + -1) = x(x + 1)

Reorder the terms:
(-2 + 5x)(x + -1) = x(x + 1)

Reorder the terms:
(-2 + 5x)(-1 + x) = x(x + 1)

Multiply (-2 + 5x) * (-1 + x)
(-2(-1 + x) + 5x * (-1 + x)) = x(x + 1)
((-1 * -2 + x * -2) + 5x * (-1 + x)) = x(x + 1)
((2 + -2x) + 5x * (-1 + x)) = x(x + 1)
(2 + -2x + (-1 * 5x + x * 5x)) = x(x + 1)
(2 + -2x + (-5x + 5x2)) = x(x + 1)

Combine like terms: -2x + -5x = -7x
(2 + -7x + 5x2) = x(x + 1)

Reorder the terms:
2 + -7x + 5x2 = x(1 + x)
2 + -7x + 5x2 = (1 * x + x * x)
2 + -7x + 5x2 = (1x + x2)

Solving
2 + -7x + 5x2 = 1x + x2

Solving for variable 'x'.

Reorder the terms:
2 + -7x + -1x + 5x2 + -1x2 = 1x + x2 + -1x + -1x2

Combine like terms: -7x + -1x = -8x
2 + -8x + 5x2 + -1x2 = 1x + x2 + -1x + -1x2

Combine like terms: 5x2 + -1x2 = 4x2
2 + -8x + 4x2 = 1x + x2 + -1x + -1x2

Reorder the terms:
2 + -8x + 4x2 = 1x + -1x + x2 + -1x2

Combine like terms: 1x + -1x = 0
2 + -8x + 4x2 = 0 + x2 + -1x2
2 + -8x + 4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
2 + -8x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + -4x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + -4x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + -4x + 2x2 = 0

Solving
1 + -4x + 2x2 = 0

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

Divide each side by '2'.
0.5 + -2x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2x + x2 = 0 + -0.5
-2x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2x + x2 = -0.5

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = -0.5 + 1

Reorder the terms:
1 + -2x + x2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + -2x + x2 = 0.5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + -1) equal to 0.707106781 and -0.707106781.

Subproblem 1
x + -1 = 0.707106781

Simplifying
x + -1 = 0.707106781

Reorder the terms:
-1 + x = 0.707106781

Solving
-1 + x = 0.707106781

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + x = 0.707106781 + 1
x = 0.707106781 + 1

Combine like terms: 0.707106781 + 1 = 1.707106781
x = 1.707106781

Simplifying
x = 1.707106781

Subproblem 2
x + -1 = -0.707106781

Simplifying
x + -1 = -0.707106781

Reorder the terms:
-1 + x = -0.707106781

Solving
-1 + x = -0.707106781

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + x = -0.707106781 + 1
x = -0.707106781 + 1

Combine like terms: -0.707106781 + 1 = 0.292893219
x = 0.292893219

Simplifying
x = 0.292893219

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.707106781, 0.292893219}
Solution
x = {1.707106781, 0.292893219}