(x-4)(x-6)=1

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Solution for (x-4)(x-6)=1 equation:


Simplifying
(x + -4)(x + -6) = 1

Reorder the terms:
(-4 + x)(x + -6) = 1

Reorder the terms:
(-4 + x)(-6 + x) = 1

Multiply (-4 + x) * (-6 + x)
(-4(-6 + x) + x(-6 + x)) = 1
((-6 * -4 + x * -4) + x(-6 + x)) = 1
((24 + -4x) + x(-6 + x)) = 1
(24 + -4x + (-6 * x + x * x)) = 1
(24 + -4x + (-6x + x2)) = 1

Combine like terms: -4x + -6x = -10x
(24 + -10x + x2) = 1

Solving
24 + -10x + x2 = 1

Solving for variable 'x'.

Reorder the terms:
24 + -1 + -10x + x2 = 1 + -1

Combine like terms: 24 + -1 = 23
23 + -10x + x2 = 1 + -1

Combine like terms: 1 + -1 = 0
23 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-23' to each side of the equation.
23 + -10x + -23 + x2 = 0 + -23

Reorder the terms:
23 + -23 + -10x + x2 = 0 + -23

Combine like terms: 23 + -23 = 0
0 + -10x + x2 = 0 + -23
-10x + x2 = 0 + -23

Combine like terms: 0 + -23 = -23
-10x + x2 = -23

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -23 + 25

Reorder the terms:
25 + -10x + x2 = -23 + 25

Combine like terms: -23 + 25 = 2
25 + -10x + x2 = 2

Factor a perfect square on the left side:
(x + -5)(x + -5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -5) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -5 = 1.414213562 Simplifying x + -5 = 1.414213562 Reorder the terms: -5 + x = 1.414213562 Solving -5 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + x = 1.414213562 + 5 x = 1.414213562 + 5 Combine like terms: 1.414213562 + 5 = 6.414213562 x = 6.414213562 Simplifying x = 6.414213562

Subproblem 2

x + -5 = -1.414213562 Simplifying x + -5 = -1.414213562 Reorder the terms: -5 + x = -1.414213562 Solving -5 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + x = -1.414213562 + 5 x = -1.414213562 + 5 Combine like terms: -1.414213562 + 5 = 3.585786438 x = 3.585786438 Simplifying x = 3.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.414213562, 3.585786438}

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