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

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


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

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

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

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

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

Solving
6 + -5x + x2 = 1

Solving for variable 'x'.

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

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

Combine like terms: 1 + -1 = 0
5 + -5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -5x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + -5x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -5x + x2 = 0 + -5
-5x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
-5x + x2 = -5

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -5 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -5 + 6.25

Combine like terms: -5 + 6.25 = 1.25
6.25 + -5x + x2 = 1.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -2.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + -2.5 = 1.118033989 Simplifying x + -2.5 = 1.118033989 Reorder the terms: -2.5 + x = 1.118033989 Solving -2.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.118033989 + 2.5 x = 1.118033989 + 2.5 Combine like terms: 1.118033989 + 2.5 = 3.618033989 x = 3.618033989 Simplifying x = 3.618033989

Subproblem 2

x + -2.5 = -1.118033989 Simplifying x + -2.5 = -1.118033989 Reorder the terms: -2.5 + x = -1.118033989 Solving -2.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.118033989 + 2.5 x = -1.118033989 + 2.5 Combine like terms: -1.118033989 + 2.5 = 1.381966011 x = 1.381966011 Simplifying x = 1.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.618033989, 1.381966011}

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