(x-1)=(x-3)(x-4)

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Solution for (x-1)=(x-3)(x-4) equation:


Simplifying
(x + -1) = (x + -3)(x + -4)

Reorder the terms:
(-1 + x) = (x + -3)(x + -4)

Remove parenthesis around (-1 + x)
-1 + x = (x + -3)(x + -4)

Reorder the terms:
-1 + x = (-3 + x)(x + -4)

Reorder the terms:
-1 + x = (-3 + x)(-4 + x)

Multiply (-3 + x) * (-4 + x)
-1 + x = (-3(-4 + x) + x(-4 + x))
-1 + x = ((-4 * -3 + x * -3) + x(-4 + x))
-1 + x = ((12 + -3x) + x(-4 + x))
-1 + x = (12 + -3x + (-4 * x + x * x))
-1 + x = (12 + -3x + (-4x + x2))

Combine like terms: -3x + -4x = -7x
-1 + x = (12 + -7x + x2)

Solving
-1 + x = 12 + -7x + x2

Solving for variable 'x'.

Reorder the terms:
-1 + -12 + x + 7x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Combine like terms: -1 + -12 = -13
-13 + x + 7x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Combine like terms: x + 7x = 8x
-13 + 8x + -1x2 = 12 + -7x + x2 + -12 + 7x + -1x2

Reorder the terms:
-13 + 8x + -1x2 = 12 + -12 + -7x + 7x + x2 + -1x2

Combine like terms: 12 + -12 = 0
-13 + 8x + -1x2 = 0 + -7x + 7x + x2 + -1x2
-13 + 8x + -1x2 = -7x + 7x + x2 + -1x2

Combine like terms: -7x + 7x = 0
-13 + 8x + -1x2 = 0 + x2 + -1x2
-13 + 8x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-13 + 8x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
13 + -8x + x2 = 0

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -8x + -13 + x2 = 0 + -13

Reorder the terms:
13 + -13 + -8x + x2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -8x + x2 = 0 + -13
-8x + x2 = 0 + -13

Combine like terms: 0 + -13 = -13
-8x + x2 = -13

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 = -13 + 16

Reorder the terms:
16 + -8x + x2 = -13 + 16

Combine like terms: -13 + 16 = 3
16 + -8x + x2 = 3

Factor a perfect square on the left side:
(x + -4)(x + -4) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -4) equal to 1.732050808 and -1.732050808.

Subproblem 1

x + -4 = 1.732050808 Simplifying x + -4 = 1.732050808 Reorder the terms: -4 + x = 1.732050808 Solving -4 + x = 1.732050808 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 = 1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x = 1.732050808 + 4 x = 1.732050808 + 4 Combine like terms: 1.732050808 + 4 = 5.732050808 x = 5.732050808 Simplifying x = 5.732050808

Subproblem 2

x + -4 = -1.732050808 Simplifying x + -4 = -1.732050808 Reorder the terms: -4 + x = -1.732050808 Solving -4 + x = -1.732050808 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 = -1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x = -1.732050808 + 4 x = -1.732050808 + 4 Combine like terms: -1.732050808 + 4 = 2.267949192 x = 2.267949192 Simplifying x = 2.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.732050808, 2.267949192}

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