5300=0.5(13x^2)+(90-x)(13x)

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Solution for 5300=0.5(13x^2)+(90-x)(13x) equation:


Simplifying
5300 = 0.5(13x2) + (90 + -1x)(13x)

Remove parenthesis around (13x2)
5300 = 0.5 * 13x2 + (90 + -1x)(13x)

Multiply 0.5 * 13
5300 = 6.5x2 + (90 + -1x)(13x)

Remove parenthesis around (13x)
5300 = 6.5x2 + (90 + -1x) * 13x

Reorder the terms for easier multiplication:
5300 = 6.5x2 + 13x(90 + -1x)
5300 = 6.5x2 + (90 * 13x + -1x * 13x)
5300 = 6.5x2 + (1170x + -13x2)

Reorder the terms:
5300 = 1170x + 6.5x2 + -13x2

Combine like terms: 6.5x2 + -13x2 = -6.5x2
5300 = 1170x + -6.5x2

Solving
5300 = 1170x + -6.5x2

Solving for variable 'x'.

Reorder the terms:
5300 + -1170x + 6.5x2 = 1170x + -1170x + -6.5x2 + 6.5x2

Combine like terms: 1170x + -1170x = 0
5300 + -1170x + 6.5x2 = 0 + -6.5x2 + 6.5x2
5300 + -1170x + 6.5x2 = -6.5x2 + 6.5x2

Combine like terms: -6.5x2 + 6.5x2 = 0.0
5300 + -1170x + 6.5x2 = 0.0

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

Divide each side by '6.5'.
815.3846154 + -180x + x2 = 0

Move the constant term to the right:

Add '-815.3846154' to each side of the equation.
815.3846154 + -180x + -815.3846154 + x2 = 0 + -815.3846154

Reorder the terms:
815.3846154 + -815.3846154 + -180x + x2 = 0 + -815.3846154

Combine like terms: 815.3846154 + -815.3846154 = 0.0000000
0.0000000 + -180x + x2 = 0 + -815.3846154
-180x + x2 = 0 + -815.3846154

Combine like terms: 0 + -815.3846154 = -815.3846154
-180x + x2 = -815.3846154

The x term is -180x.  Take half its coefficient (-90).
Square it (8100) and add it to both sides.

Add '8100' to each side of the equation.
-180x + 8100 + x2 = -815.3846154 + 8100

Reorder the terms:
8100 + -180x + x2 = -815.3846154 + 8100

Combine like terms: -815.3846154 + 8100 = 7284.6153846
8100 + -180x + x2 = 7284.6153846

Factor a perfect square on the left side:
(x + -90)(x + -90) = 7284.6153846

Calculate the square root of the right side: 85.349958316

Break this problem into two subproblems by setting 
(x + -90) equal to 85.349958316 and -85.349958316.

Subproblem 1

x + -90 = 85.349958316 Simplifying x + -90 = 85.349958316 Reorder the terms: -90 + x = 85.349958316 Solving -90 + x = 85.349958316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = 85.349958316 + 90 Combine like terms: -90 + 90 = 0 0 + x = 85.349958316 + 90 x = 85.349958316 + 90 Combine like terms: 85.349958316 + 90 = 175.349958316 x = 175.349958316 Simplifying x = 175.349958316

Subproblem 2

x + -90 = -85.349958316 Simplifying x + -90 = -85.349958316 Reorder the terms: -90 + x = -85.349958316 Solving -90 + x = -85.349958316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = -85.349958316 + 90 Combine like terms: -90 + 90 = 0 0 + x = -85.349958316 + 90 x = -85.349958316 + 90 Combine like terms: -85.349958316 + 90 = 4.650041684 x = 4.650041684 Simplifying x = 4.650041684

Solution

The solution to the problem is based on the solutions from the subproblems. x = {175.349958316, 4.650041684}

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