(5x)(91-2x)=180

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Solution for (5x)(91-2x)=180 equation: 


Simplifying
(5x)(91 + -2x) = 180

Remove parenthesis around (5x)
5x(91 + -2x) = 180
(91 * 5x + -2x * 5x) = 180
(455x + -10x2) = 180

Solving
455x + -10x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 455x + -10x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 455x + -10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-36 + 91x + -2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-36 + 91x + -2x2)' equal to zero and attempt to solve:

Simplifying
-36 + 91x + -2x2 = 0

Solving
-36 + 91x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
18 + -45.5x + x2 = 0

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -45.5x + -18 + x2 = 0 + -18

Reorder the terms:
18 + -18 + -45.5x + x2 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -45.5x + x2 = 0 + -18
-45.5x + x2 = 0 + -18

Combine like terms: 0 + -18 = -18
-45.5x + x2 = -18

The x term is -45.5x.  Take half its coefficient (-22.75).
Square it (517.5625) and add it to both sides.

Add '517.5625' to each side of the equation.
-45.5x + 517.5625 + x2 = -18 + 517.5625

Reorder the terms:
517.5625 + -45.5x + x2 = -18 + 517.5625

Combine like terms: -18 + 517.5625 = 499.5625
517.5625 + -45.5x + x2 = 499.5625

Factor a perfect square on the left side:
(x + -22.75)(x + -22.75) = 499.5625

Calculate the square root of the right side: 22.350894837

Break this problem into two subproblems by setting 
(x + -22.75) equal to 22.350894837 and -22.350894837.


Subproblem 1
x + -22.75 = 22.350894837

Simplifying
x + -22.75 = 22.350894837

Reorder the terms:
-22.75 + x = 22.350894837

Solving
-22.75 + x = 22.350894837

Solving for variable 'x'.

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

Add '22.75' to each side of the equation.
-22.75 + 22.75 + x = 22.350894837 + 22.75

Combine like terms: -22.75 + 22.75 = 0.00
0.00 + x = 22.350894837 + 22.75
x = 22.350894837 + 22.75

Combine like terms: 22.350894837 + 22.75 = 45.100894837
x = 45.100894837

Simplifying
x = 45.100894837


Subproblem 2
x + -22.75 = -22.350894837

Simplifying
x + -22.75 = -22.350894837

Reorder the terms:
-22.75 + x = -22.350894837

Solving
-22.75 + x = -22.350894837

Solving for variable 'x'.

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

Add '22.75' to each side of the equation.
-22.75 + 22.75 + x = -22.350894837 + 22.75

Combine like terms: -22.75 + 22.75 = 0.00
0.00 + x = -22.350894837 + 22.75
x = -22.350894837 + 22.75

Combine like terms: -22.350894837 + 22.75 = 0.399105163
x = 0.399105163

Simplifying
x = 0.399105163


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {45.100894837, 0.399105163}
Solution
x = {45.100894837, 0.399105163}

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