(6-x)(x-4)=(6-x)(-2+0.5x)

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Solution for (6-x)(x-4)=(6-x)(-2+0.5x) equation:


Simplifying
(6 + -1x)(x + -4) = (6 + -1x)(-2 + 0.5x)

Reorder the terms:
(6 + -1x)(-4 + x) = (6 + -1x)(-2 + 0.5x)

Multiply (6 + -1x) * (-4 + x)
(6(-4 + x) + -1x * (-4 + x)) = (6 + -1x)(-2 + 0.5x)
((-4 * 6 + x * 6) + -1x * (-4 + x)) = (6 + -1x)(-2 + 0.5x)
((-24 + 6x) + -1x * (-4 + x)) = (6 + -1x)(-2 + 0.5x)
(-24 + 6x + (-4 * -1x + x * -1x)) = (6 + -1x)(-2 + 0.5x)
(-24 + 6x + (4x + -1x2)) = (6 + -1x)(-2 + 0.5x)

Combine like terms: 6x + 4x = 10x
(-24 + 10x + -1x2) = (6 + -1x)(-2 + 0.5x)

Multiply (6 + -1x) * (-2 + 0.5x)
-24 + 10x + -1x2 = (6(-2 + 0.5x) + -1x * (-2 + 0.5x))
-24 + 10x + -1x2 = ((-2 * 6 + 0.5x * 6) + -1x * (-2 + 0.5x))
-24 + 10x + -1x2 = ((-12 + 3x) + -1x * (-2 + 0.5x))
-24 + 10x + -1x2 = (-12 + 3x + (-2 * -1x + 0.5x * -1x))
-24 + 10x + -1x2 = (-12 + 3x + (2x + -0.5x2))

Combine like terms: 3x + 2x = 5x
-24 + 10x + -1x2 = (-12 + 5x + -0.5x2)

Solving
-24 + 10x + -1x2 = -12 + 5x + -0.5x2

Solving for variable 'x'.

Reorder the terms:
-24 + 12 + 10x + -5x + -1x2 + 0.5x2 = -12 + 5x + -0.5x2 + 12 + -5x + 0.5x2

Combine like terms: -24 + 12 = -12
-12 + 10x + -5x + -1x2 + 0.5x2 = -12 + 5x + -0.5x2 + 12 + -5x + 0.5x2

Combine like terms: 10x + -5x = 5x
-12 + 5x + -1x2 + 0.5x2 = -12 + 5x + -0.5x2 + 12 + -5x + 0.5x2

Combine like terms: -1x2 + 0.5x2 = -0.5x2
-12 + 5x + -0.5x2 = -12 + 5x + -0.5x2 + 12 + -5x + 0.5x2

Reorder the terms:
-12 + 5x + -0.5x2 = -12 + 12 + 5x + -5x + -0.5x2 + 0.5x2

Combine like terms: -12 + 12 = 0
-12 + 5x + -0.5x2 = 0 + 5x + -5x + -0.5x2 + 0.5x2
-12 + 5x + -0.5x2 = 5x + -5x + -0.5x2 + 0.5x2

Combine like terms: 5x + -5x = 0
-12 + 5x + -0.5x2 = 0 + -0.5x2 + 0.5x2
-12 + 5x + -0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
-12 + 5x + -0.5x2 = 0.0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
24 + -10x + x2 = 0

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + -10x + -24 + x2 = 0 + -24

Reorder the terms:
24 + -24 + -10x + x2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + -10x + x2 = 0 + -24
-10x + x2 = 0 + -24

Combine like terms: 0 + -24 = -24
-10x + x2 = -24

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 = -24 + 25

Reorder the terms:
25 + -10x + x2 = -24 + 25

Combine like terms: -24 + 25 = 1
25 + -10x + x2 = 1

Factor a perfect square on the left side:
(x + -5)(x + -5) = 1

Calculate the square root of the right side: 1

Break this problem into two subproblems by setting 
(x + -5) equal to 1 and -1.

Subproblem 1

x + -5 = 1 Simplifying x + -5 = 1 Reorder the terms: -5 + x = 1 Solving -5 + x = 1 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 + 5 Combine like terms: -5 + 5 = 0 0 + x = 1 + 5 x = 1 + 5 Combine like terms: 1 + 5 = 6 x = 6 Simplifying x = 6

Subproblem 2

x + -5 = -1 Simplifying x + -5 = -1 Reorder the terms: -5 + x = -1 Solving -5 + x = -1 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 + 5 Combine like terms: -5 + 5 = 0 0 + x = -1 + 5 x = -1 + 5 Combine like terms: -1 + 5 = 4 x = 4 Simplifying x = 4

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6, 4}

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