(x^2-5x+6)(x-1)=(x^2+8x+16)(x-4)+7

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Solution for (x^2-5x+6)(x-1)=(x^2+8x+16)(x-4)+7 equation:


Simplifying
(x2 + -5x + 6)(x + -1) = (x2 + 8x + 16)(x + -4) + 7

Reorder the terms:
(6 + -5x + x2)(x + -1) = (x2 + 8x + 16)(x + -4) + 7

Reorder the terms:
(6 + -5x + x2)(-1 + x) = (x2 + 8x + 16)(x + -4) + 7

Multiply (6 + -5x + x2) * (-1 + x)
(6(-1 + x) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7
((-1 * 6 + x * 6) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7
((-6 + 6x) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7
(-6 + 6x + (-1 * -5x + x * -5x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7
(-6 + 6x + (5x + -5x2) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7
(-6 + 6x + 5x + -5x2 + (-1 * x2 + x * x2)) = (x2 + 8x + 16)(x + -4) + 7
(-6 + 6x + 5x + -5x2 + (-1x2 + x3)) = (x2 + 8x + 16)(x + -4) + 7

Combine like terms: 6x + 5x = 11x
(-6 + 11x + -5x2 + -1x2 + x3) = (x2 + 8x + 16)(x + -4) + 7

Combine like terms: -5x2 + -1x2 = -6x2
(-6 + 11x + -6x2 + x3) = (x2 + 8x + 16)(x + -4) + 7

Reorder the terms:
-6 + 11x + -6x2 + x3 = (16 + 8x + x2)(x + -4) + 7

Reorder the terms:
-6 + 11x + -6x2 + x3 = (16 + 8x + x2)(-4 + x) + 7

Multiply (16 + 8x + x2) * (-4 + x)
-6 + 11x + -6x2 + x3 = (16(-4 + x) + 8x * (-4 + x) + x2(-4 + x)) + 7
-6 + 11x + -6x2 + x3 = ((-4 * 16 + x * 16) + 8x * (-4 + x) + x2(-4 + x)) + 7
-6 + 11x + -6x2 + x3 = ((-64 + 16x) + 8x * (-4 + x) + x2(-4 + x)) + 7
-6 + 11x + -6x2 + x3 = (-64 + 16x + (-4 * 8x + x * 8x) + x2(-4 + x)) + 7
-6 + 11x + -6x2 + x3 = (-64 + 16x + (-32x + 8x2) + x2(-4 + x)) + 7
-6 + 11x + -6x2 + x3 = (-64 + 16x + -32x + 8x2 + (-4 * x2 + x * x2)) + 7
-6 + 11x + -6x2 + x3 = (-64 + 16x + -32x + 8x2 + (-4x2 + x3)) + 7

Combine like terms: 16x + -32x = -16x
-6 + 11x + -6x2 + x3 = (-64 + -16x + 8x2 + -4x2 + x3) + 7

Combine like terms: 8x2 + -4x2 = 4x2
-6 + 11x + -6x2 + x3 = (-64 + -16x + 4x2 + x3) + 7

Reorder the terms:
-6 + 11x + -6x2 + x3 = -64 + 7 + -16x + 4x2 + x3

Combine like terms: -64 + 7 = -57
-6 + 11x + -6x2 + x3 = -57 + -16x + 4x2 + x3

Add '-1x3' to each side of the equation.
-6 + 11x + -6x2 + x3 + -1x3 = -57 + -16x + 4x2 + x3 + -1x3

Combine like terms: x3 + -1x3 = 0
-6 + 11x + -6x2 + 0 = -57 + -16x + 4x2 + x3 + -1x3
-6 + 11x + -6x2 = -57 + -16x + 4x2 + x3 + -1x3

Combine like terms: x3 + -1x3 = 0
-6 + 11x + -6x2 = -57 + -16x + 4x2 + 0
-6 + 11x + -6x2 = -57 + -16x + 4x2

Solving
-6 + 11x + -6x2 = -57 + -16x + 4x2

Solving for variable 'x'.

Reorder the terms:
-6 + 57 + 11x + 16x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2

Combine like terms: -6 + 57 = 51
51 + 11x + 16x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2

Combine like terms: 11x + 16x = 27x
51 + 27x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2

Combine like terms: -6x2 + -4x2 = -10x2
51 + 27x + -10x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2

Reorder the terms:
51 + 27x + -10x2 = -57 + 57 + -16x + 16x + 4x2 + -4x2

Combine like terms: -57 + 57 = 0
51 + 27x + -10x2 = 0 + -16x + 16x + 4x2 + -4x2
51 + 27x + -10x2 = -16x + 16x + 4x2 + -4x2

Combine like terms: -16x + 16x = 0
51 + 27x + -10x2 = 0 + 4x2 + -4x2
51 + 27x + -10x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
51 + 27x + -10x2 = 0

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
-5.1 + -2.7x + x2 = 0.0

Move the constant term to the right:

Add '5.1' to each side of the equation.
-5.1 + -2.7x + 5.1 + x2 = 0.0 + 5.1

Reorder the terms:
-5.1 + 5.1 + -2.7x + x2 = 0.0 + 5.1

Combine like terms: -5.1 + 5.1 = 0.0
0.0 + -2.7x + x2 = 0.0 + 5.1
-2.7x + x2 = 0.0 + 5.1

Combine like terms: 0.0 + 5.1 = 5.1
-2.7x + x2 = 5.1

The x term is -2.7x.  Take half its coefficient (-1.35).
Square it (1.8225) and add it to both sides.

Add '1.8225' to each side of the equation.
-2.7x + 1.8225 + x2 = 5.1 + 1.8225

Reorder the terms:
1.8225 + -2.7x + x2 = 5.1 + 1.8225

Combine like terms: 5.1 + 1.8225 = 6.9225
1.8225 + -2.7x + x2 = 6.9225

Factor a perfect square on the left side:
(x + -1.35)(x + -1.35) = 6.9225

Calculate the square root of the right side: 2.631064423

Break this problem into two subproblems by setting 
(x + -1.35) equal to 2.631064423 and -2.631064423.

Subproblem 1

x + -1.35 = 2.631064423 Simplifying x + -1.35 = 2.631064423 Reorder the terms: -1.35 + x = 2.631064423 Solving -1.35 + x = 2.631064423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = 2.631064423 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = 2.631064423 + 1.35 x = 2.631064423 + 1.35 Combine like terms: 2.631064423 + 1.35 = 3.981064423 x = 3.981064423 Simplifying x = 3.981064423

Subproblem 2

x + -1.35 = -2.631064423 Simplifying x + -1.35 = -2.631064423 Reorder the terms: -1.35 + x = -2.631064423 Solving -1.35 + x = -2.631064423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = -2.631064423 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = -2.631064423 + 1.35 x = -2.631064423 + 1.35 Combine like terms: -2.631064423 + 1.35 = -1.281064423 x = -1.281064423 Simplifying x = -1.281064423

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.981064423, -1.281064423}

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