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

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


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

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

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

Multiply (-3 + x) * (-2 + x)
(-3(-2 + x) + x(-2 + x)) = (x + -2) * 2 + -1
((-2 * -3 + x * -3) + x(-2 + x)) = (x + -2) * 2 + -1
((6 + -3x) + x(-2 + x)) = (x + -2) * 2 + -1
(6 + -3x + (-2 * x + x * x)) = (x + -2) * 2 + -1
(6 + -3x + (-2x + x2)) = (x + -2) * 2 + -1

Combine like terms: -3x + -2x = -5x
(6 + -5x + x2) = (x + -2) * 2 + -1

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

Reorder the terms for easier multiplication:
6 + -5x + x2 = 2(-2 + x) + -1
6 + -5x + x2 = (-2 * 2 + x * 2) + -1
6 + -5x + x2 = (-4 + 2x) + -1

Reorder the terms:
6 + -5x + x2 = -4 + -1 + 2x

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

Solving
6 + -5x + x2 = -5 + 2x

Solving for variable 'x'.

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

Combine like terms: 6 + 5 = 11
11 + -5x + -2x + x2 = -5 + 2x + 5 + -2x

Combine like terms: -5x + -2x = -7x
11 + -7x + x2 = -5 + 2x + 5 + -2x

Reorder the terms:
11 + -7x + x2 = -5 + 5 + 2x + -2x

Combine like terms: -5 + 5 = 0
11 + -7x + x2 = 0 + 2x + -2x
11 + -7x + x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
11 + -7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-11' to each side of the equation.
11 + -7x + -11 + x2 = 0 + -11

Reorder the terms:
11 + -11 + -7x + x2 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -7x + x2 = 0 + -11
-7x + x2 = 0 + -11

Combine like terms: 0 + -11 = -11
-7x + x2 = -11

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -11 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -11 + 12.25

Combine like terms: -11 + 12.25 = 1.25
12.25 + -7x + x2 = 1.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + -3.5 = 1.118033989 Simplifying x + -3.5 = 1.118033989 Reorder the terms: -3.5 + x = 1.118033989 Solving -3.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 1.118033989 + 3.5 x = 1.118033989 + 3.5 Combine like terms: 1.118033989 + 3.5 = 4.618033989 x = 4.618033989 Simplifying x = 4.618033989

Subproblem 2

x + -3.5 = -1.118033989 Simplifying x + -3.5 = -1.118033989 Reorder the terms: -3.5 + x = -1.118033989 Solving -3.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -1.118033989 + 3.5 x = -1.118033989 + 3.5 Combine like terms: -1.118033989 + 3.5 = 2.381966011 x = 2.381966011 Simplifying x = 2.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.618033989, 2.381966011}

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