(x-5)(x-1)=1

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


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

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

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

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

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

Solving
5 + -6x + x2 = 1

Solving for variable 'x'.

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -6x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + -6x + x2 = 0 + -4

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

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

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = -4 + 9

Reorder the terms:
9 + -6x + x2 = -4 + 9

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

Factor a perfect square on the left side:
(x + -3)(x + -3) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + -3) equal to 2.236067978 and -2.236067978.

Subproblem 1

x + -3 = 2.236067978 Simplifying x + -3 = 2.236067978 Reorder the terms: -3 + x = 2.236067978 Solving -3 + x = 2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + x = 2.236067978 + 3 x = 2.236067978 + 3 Combine like terms: 2.236067978 + 3 = 5.236067978 x = 5.236067978 Simplifying x = 5.236067978

Subproblem 2

x + -3 = -2.236067978 Simplifying x + -3 = -2.236067978 Reorder the terms: -3 + x = -2.236067978 Solving -3 + x = -2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + x = -2.236067978 + 3 x = -2.236067978 + 3 Combine like terms: -2.236067978 + 3 = 0.763932022 x = 0.763932022 Simplifying x = 0.763932022

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.236067978, 0.763932022}

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