(x-5)(x).5=42

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Solution for (x-5)(x).5=42 equation: 


Simplifying
(x + -5)(x) * 0.5 = 42

Reorder the terms:
(-5 + x)(x) * 0.5 = 42

Reorder the terms for easier multiplication:
0.5x(-5 + x) = 42
(-5 * 0.5x + x * 0.5x) = 42
(-2.5x + 0.5x2) = 42

Solving
-2.5x + 0.5x2 = 42

Solving for variable 'x'.

Reorder the terms:
-42 + -2.5x + 0.5x2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + -2.5x + 0.5x2 = 0

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

Divide each side by '0.5'.
-84 + -5x + x2 = 0

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -5x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + -5x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -5x + x2 = 0 + 84
-5x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
-5x + x2 = 84

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 84 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 84 + 6.25

Combine like terms: 84 + 6.25 = 90.25
6.25 + -5x + x2 = 90.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 90.25

Calculate the square root of the right side: 9.5

Break this problem into two subproblems by setting 
(x + -2.5) equal to 9.5 and -9.5.


Subproblem 1
x + -2.5 = 9.5

Simplifying
x + -2.5 = 9.5

Reorder the terms:
-2.5 + x = 9.5

Solving
-2.5 + x = 9.5

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 9.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 9.5 + 2.5
x = 9.5 + 2.5

Combine like terms: 9.5 + 2.5 = 12
x = 12

Simplifying
x = 12


Subproblem 2
x + -2.5 = -9.5

Simplifying
x + -2.5 = -9.5

Reorder the terms:
-2.5 + x = -9.5

Solving
-2.5 + x = -9.5

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -9.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -9.5 + 2.5
x = -9.5 + 2.5

Combine like terms: -9.5 + 2.5 = -7
x = -7

Simplifying
x = -7


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12, -7}

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