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

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


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

Reorder the terms:
1 + 2x(-2 + x) = 5(x + -1)
1 + (-2 * 2x + x * 2x) = 5(x + -1)
1 + (-4x + 2x2) = 5(x + -1)

Reorder the terms:
1 + -4x + 2x2 = 5(-1 + x)
1 + -4x + 2x2 = (-1 * 5 + x * 5)
1 + -4x + 2x2 = (-5 + 5x)

Solving
1 + -4x + 2x2 = -5 + 5x

Solving for variable 'x'.

Reorder the terms:
1 + 5 + -4x + -5x + 2x2 = -5 + 5x + 5 + -5x

Combine like terms: 1 + 5 = 6
6 + -4x + -5x + 2x2 = -5 + 5x + 5 + -5x

Combine like terms: -4x + -5x = -9x
6 + -9x + 2x2 = -5 + 5x + 5 + -5x

Reorder the terms:
6 + -9x + 2x2 = -5 + 5 + 5x + -5x

Combine like terms: -5 + 5 = 0
6 + -9x + 2x2 = 0 + 5x + -5x
6 + -9x + 2x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
6 + -9x + 2x2 = 0

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

Divide each side by '2'.
3 + -4.5x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.5x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.5x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.5x + x2 = 0 + -3
-4.5x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.5x + x2 = -3

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = -3 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = -3 + 5.0625

Combine like terms: -3 + 5.0625 = 2.0625
5.0625 + -4.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + -2.25) equal to 1.436140662 and -1.436140662.

Subproblem 1

x + -2.25 = 1.436140662 Simplifying x + -2.25 = 1.436140662 Reorder the terms: -2.25 + x = 1.436140662 Solving -2.25 + x = 1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 1.436140662 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 1.436140662 + 2.25 x = 1.436140662 + 2.25 Combine like terms: 1.436140662 + 2.25 = 3.686140662 x = 3.686140662 Simplifying x = 3.686140662

Subproblem 2

x + -2.25 = -1.436140662 Simplifying x + -2.25 = -1.436140662 Reorder the terms: -2.25 + x = -1.436140662 Solving -2.25 + x = -1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -1.436140662 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -1.436140662 + 2.25 x = -1.436140662 + 2.25 Combine like terms: -1.436140662 + 2.25 = 0.813859338 x = 0.813859338 Simplifying x = 0.813859338

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.686140662, 0.813859338}

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