.5(x-20)(x)=78

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Solution for .5(x-20)(x)=78 equation: 


Simplifying
0.5(x + -20)(x) = 78

Reorder the terms:
0.5(-20 + x)(x) = 78

Reorder the terms for easier multiplication:
0.5x(-20 + x) = 78
(-20 * 0.5x + x * 0.5x) = 78
(-10x + 0.5x2) = 78

Solving
-10x + 0.5x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + -10x + 0.5x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + -10x + 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'.
-156 + -20x + x2 = 0

Move the constant term to the right:

Add '156' to each side of the equation.
-156 + -20x + 156 + x2 = 0 + 156

Reorder the terms:
-156 + 156 + -20x + x2 = 0 + 156

Combine like terms: -156 + 156 = 0
0 + -20x + x2 = 0 + 156
-20x + x2 = 0 + 156

Combine like terms: 0 + 156 = 156
-20x + x2 = 156

The x term is -20x.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20x + 100 + x2 = 156 + 100

Reorder the terms:
100 + -20x + x2 = 156 + 100

Combine like terms: 156 + 100 = 256
100 + -20x + x2 = 256

Factor a perfect square on the left side:
(x + -10)(x + -10) = 256

Calculate the square root of the right side: 16

Break this problem into two subproblems by setting 
(x + -10) equal to 16 and -16.


Subproblem 1
x + -10 = 16

Simplifying
x + -10 = 16

Reorder the terms:
-10 + x = 16

Solving
-10 + x = 16

Solving for variable 'x'.

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

Add '10' to each side of the equation.
-10 + 10 + x = 16 + 10

Combine like terms: -10 + 10 = 0
0 + x = 16 + 10
x = 16 + 10

Combine like terms: 16 + 10 = 26
x = 26

Simplifying
x = 26


Subproblem 2
x + -10 = -16

Simplifying
x + -10 = -16

Reorder the terms:
-10 + x = -16

Solving
-10 + x = -16

Solving for variable 'x'.

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

Add '10' to each side of the equation.
-10 + 10 + x = -16 + 10

Combine like terms: -10 + 10 = 0
0 + x = -16 + 10
x = -16 + 10

Combine like terms: -16 + 10 = -6
x = -6

Simplifying
x = -6


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {26, -6}

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