(2x-1)2=(x-1)(x-3)+1

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Solution for (2x-1)2=(x-1)(x-3)+1 equation:


Simplifying
(2x + -1) * 2 = (x + -1)(x + -3) + 1

Reorder the terms:
(-1 + 2x) * 2 = (x + -1)(x + -3) + 1

Reorder the terms for easier multiplication:
2(-1 + 2x) = (x + -1)(x + -3) + 1
(-1 * 2 + 2x * 2) = (x + -1)(x + -3) + 1
(-2 + 4x) = (x + -1)(x + -3) + 1

Reorder the terms:
-2 + 4x = (-1 + x)(x + -3) + 1

Reorder the terms:
-2 + 4x = (-1 + x)(-3 + x) + 1

Multiply (-1 + x) * (-3 + x)
-2 + 4x = (-1(-3 + x) + x(-3 + x)) + 1
-2 + 4x = ((-3 * -1 + x * -1) + x(-3 + x)) + 1
-2 + 4x = ((3 + -1x) + x(-3 + x)) + 1
-2 + 4x = (3 + -1x + (-3 * x + x * x)) + 1
-2 + 4x = (3 + -1x + (-3x + x2)) + 1

Combine like terms: -1x + -3x = -4x
-2 + 4x = (3 + -4x + x2) + 1

Reorder the terms:
-2 + 4x = 3 + 1 + -4x + x2

Combine like terms: 3 + 1 = 4
-2 + 4x = 4 + -4x + x2

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

Solving for variable 'x'.

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

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

Combine like terms: 4x + 4x = 8x
-6 + 8x + -1x2 = 4 + -4x + x2 + -4 + 4x + -1x2

Reorder the terms:
-6 + 8x + -1x2 = 4 + -4 + -4x + 4x + x2 + -1x2

Combine like terms: 4 + -4 = 0
-6 + 8x + -1x2 = 0 + -4x + 4x + x2 + -1x2
-6 + 8x + -1x2 = -4x + 4x + x2 + -1x2

Combine like terms: -4x + 4x = 0
-6 + 8x + -1x2 = 0 + x2 + -1x2
-6 + 8x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-6 + 8x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
6 + -8x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -8x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + -8x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -8x + x2 = 0 + -6
-8x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
-8x + x2 = -6

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = -6 + 16

Reorder the terms:
16 + -8x + x2 = -6 + 16

Combine like terms: -6 + 16 = 10
16 + -8x + x2 = 10

Factor a perfect square on the left side:
(x + -4)(x + -4) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x + -4) equal to 3.16227766 and -3.16227766.

Subproblem 1

x + -4 = 3.16227766 Simplifying x + -4 = 3.16227766 Reorder the terms: -4 + x = 3.16227766 Solving -4 + x = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + x = 3.16227766 + 4 x = 3.16227766 + 4 Combine like terms: 3.16227766 + 4 = 7.16227766 x = 7.16227766 Simplifying x = 7.16227766

Subproblem 2

x + -4 = -3.16227766 Simplifying x + -4 = -3.16227766 Reorder the terms: -4 + x = -3.16227766 Solving -4 + x = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + x = -3.16227766 + 4 x = -3.16227766 + 4 Combine like terms: -3.16227766 + 4 = 0.83772234 x = 0.83772234 Simplifying x = 0.83772234

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.16227766, 0.83772234}

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