x(x)+(x+13)(x+13)=90000

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Solution for x(x)+(x+13)(x+13)=90000 equation:


Simplifying
x(x) + (x + 13)(x + 13) = 90000

Multiply x * x
x2 + (x + 13)(x + 13) = 90000

Reorder the terms:
x2 + (13 + x)(x + 13) = 90000

Reorder the terms:
x2 + (13 + x)(13 + x) = 90000

Multiply (13 + x) * (13 + x)
x2 + (13(13 + x) + x(13 + x)) = 90000
x2 + ((13 * 13 + x * 13) + x(13 + x)) = 90000
x2 + ((169 + 13x) + x(13 + x)) = 90000
x2 + (169 + 13x + (13 * x + x * x)) = 90000
x2 + (169 + 13x + (13x + x2)) = 90000

Combine like terms: 13x + 13x = 26x
x2 + (169 + 26x + x2) = 90000

Reorder the terms:
169 + 26x + x2 + x2 = 90000

Combine like terms: x2 + x2 = 2x2
169 + 26x + 2x2 = 90000

Solving
169 + 26x + 2x2 = 90000

Solving for variable 'x'.

Reorder the terms:
169 + -90000 + 26x + 2x2 = 90000 + -90000

Combine like terms: 169 + -90000 = -89831
-89831 + 26x + 2x2 = 90000 + -90000

Combine like terms: 90000 + -90000 = 0
-89831 + 26x + 2x2 = 0

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

Divide each side by '2'.
-44915.5 + 13x + x2 = 0

Move the constant term to the right:

Add '44915.5' to each side of the equation.
-44915.5 + 13x + 44915.5 + x2 = 0 + 44915.5

Reorder the terms:
-44915.5 + 44915.5 + 13x + x2 = 0 + 44915.5

Combine like terms: -44915.5 + 44915.5 = 0.0
0.0 + 13x + x2 = 0 + 44915.5
13x + x2 = 0 + 44915.5

Combine like terms: 0 + 44915.5 = 44915.5
13x + x2 = 44915.5

The x term is 13x.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13x + 42.25 + x2 = 44915.5 + 42.25

Reorder the terms:
42.25 + 13x + x2 = 44915.5 + 42.25

Combine like terms: 44915.5 + 42.25 = 44957.75
42.25 + 13x + x2 = 44957.75

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 44957.75

Calculate the square root of the right side: 212.032426765

Break this problem into two subproblems by setting 
(x + 6.5) equal to 212.032426765 and -212.032426765.

Subproblem 1

x + 6.5 = 212.032426765 Simplifying x + 6.5 = 212.032426765 Reorder the terms: 6.5 + x = 212.032426765 Solving 6.5 + x = 212.032426765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = 212.032426765 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 212.032426765 + -6.5 x = 212.032426765 + -6.5 Combine like terms: 212.032426765 + -6.5 = 205.532426765 x = 205.532426765 Simplifying x = 205.532426765

Subproblem 2

x + 6.5 = -212.032426765 Simplifying x + 6.5 = -212.032426765 Reorder the terms: 6.5 + x = -212.032426765 Solving 6.5 + x = -212.032426765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = -212.032426765 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -212.032426765 + -6.5 x = -212.032426765 + -6.5 Combine like terms: -212.032426765 + -6.5 = -218.532426765 x = -218.532426765 Simplifying x = -218.532426765

Solution

The solution to the problem is based on the solutions from the subproblems. x = {205.532426765, -218.532426765}

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