(x-3)(x-5)=1

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Solution for (x-3)(x-5)=1 equation:


Simplifying
(x + -3)(x + -5) = 1

Reorder the terms:
(-3 + x)(x + -5) = 1

Reorder the terms:
(-3 + x)(-5 + x) = 1

Multiply (-3 + x) * (-5 + x)
(-3(-5 + x) + x(-5 + x)) = 1
((-5 * -3 + x * -3) + x(-5 + x)) = 1
((15 + -3x) + x(-5 + x)) = 1
(15 + -3x + (-5 * x + x * x)) = 1
(15 + -3x + (-5x + x2)) = 1

Combine like terms: -3x + -5x = -8x
(15 + -8x + x2) = 1

Solving
15 + -8x + x2 = 1

Solving for variable 'x'.

Reorder the terms:
15 + -1 + -8x + x2 = 1 + -1

Combine like terms: 15 + -1 = 14
14 + -8x + x2 = 1 + -1

Combine like terms: 1 + -1 = 0
14 + -8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-14' to each side of the equation.
14 + -8x + -14 + x2 = 0 + -14

Reorder the terms:
14 + -14 + -8x + x2 = 0 + -14

Combine like terms: 14 + -14 = 0
0 + -8x + x2 = 0 + -14
-8x + x2 = 0 + -14

Combine like terms: 0 + -14 = -14
-8x + x2 = -14

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 = -14 + 16

Reorder the terms:
16 + -8x + x2 = -14 + 16

Combine like terms: -14 + 16 = 2
16 + -8x + x2 = 2

Factor a perfect square on the left side:
(x + -4)(x + -4) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -4) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -4 = 1.414213562 Simplifying x + -4 = 1.414213562 Reorder the terms: -4 + x = 1.414213562 Solving -4 + x = 1.414213562 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 = 1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + x = 1.414213562 + 4 x = 1.414213562 + 4 Combine like terms: 1.414213562 + 4 = 5.414213562 x = 5.414213562 Simplifying x = 5.414213562

Subproblem 2

x + -4 = -1.414213562 Simplifying x + -4 = -1.414213562 Reorder the terms: -4 + x = -1.414213562 Solving -4 + x = -1.414213562 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 = -1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + x = -1.414213562 + 4 x = -1.414213562 + 4 Combine like terms: -1.414213562 + 4 = 2.585786438 x = 2.585786438 Simplifying x = 2.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.414213562, 2.585786438}

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