x^2-6x=(.5x+42)

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Solution for x^2-6x=(.5x+42) equation:


Simplifying
x2 + -6x = (0.5x + 42)

Reorder the terms:
-6x + x2 = (0.5x + 42)

Reorder the terms:
-6x + x2 = (42 + 0.5x)

Remove parenthesis around (42 + 0.5x)
-6x + x2 = 42 + 0.5x

Solving
-6x + x2 = 42 + 0.5x

Solving for variable 'x'.

Reorder the terms:
-42 + -6x + -0.5x + x2 = 42 + 0.5x + -42 + -0.5x

Combine like terms: -6x + -0.5x = -6.5x
-42 + -6.5x + x2 = 42 + 0.5x + -42 + -0.5x

Reorder the terms:
-42 + -6.5x + x2 = 42 + -42 + 0.5x + -0.5x

Combine like terms: 42 + -42 = 0
-42 + -6.5x + x2 = 0 + 0.5x + -0.5x
-42 + -6.5x + x2 = 0.5x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
-42 + -6.5x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + -6.5x + 42 + x2 = 0.0 + 42

Reorder the terms:
-42 + 42 + -6.5x + x2 = 0.0 + 42

Combine like terms: -42 + 42 = 0
0 + -6.5x + x2 = 0.0 + 42
-6.5x + x2 = 0.0 + 42

Combine like terms: 0.0 + 42 = 42
-6.5x + x2 = 42

The x term is -6.5x.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5x + 10.5625 + x2 = 42 + 10.5625

Reorder the terms:
10.5625 + -6.5x + x2 = 42 + 10.5625

Combine like terms: 42 + 10.5625 = 52.5625
10.5625 + -6.5x + x2 = 52.5625

Factor a perfect square on the left side:
(x + -3.25)(x + -3.25) = 52.5625

Calculate the square root of the right side: 7.25

Break this problem into two subproblems by setting 
(x + -3.25) equal to 7.25 and -7.25.

Subproblem 1

x + -3.25 = 7.25 Simplifying x + -3.25 = 7.25 Reorder the terms: -3.25 + x = 7.25 Solving -3.25 + x = 7.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = 7.25 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = 7.25 + 3.25 x = 7.25 + 3.25 Combine like terms: 7.25 + 3.25 = 10.5 x = 10.5 Simplifying x = 10.5

Subproblem 2

x + -3.25 = -7.25 Simplifying x + -3.25 = -7.25 Reorder the terms: -3.25 + x = -7.25 Solving -3.25 + x = -7.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = -7.25 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = -7.25 + 3.25 x = -7.25 + 3.25 Combine like terms: -7.25 + 3.25 = -4 x = -4 Simplifying x = -4

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.5, -4}

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