x(x+13)=40

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Solution for x(x+13)=40 equation: 


Simplifying
x(x + 13) = 40

Reorder the terms:
x(13 + x) = 40
(13 * x + x * x) = 40
(13x + x2) = 40

Solving
13x + x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + 13x + x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 13x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 13x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + 13x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 13x + x2 = 0 + 40
13x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
13x + x2 = 40

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 = 40 + 42.25

Reorder the terms:
42.25 + 13x + x2 = 40 + 42.25

Combine like terms: 40 + 42.25 = 82.25
42.25 + 13x + x2 = 82.25

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(x + 6.5) equal to 9.069178574 and -9.069178574.


Subproblem 1
x + 6.5 = 9.069178574

Simplifying
x + 6.5 = 9.069178574

Reorder the terms:
6.5 + x = 9.069178574

Solving
6.5 + x = 9.069178574

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 = 9.069178574 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = 9.069178574 + -6.5
x = 9.069178574 + -6.5

Combine like terms: 9.069178574 + -6.5 = 2.569178574
x = 2.569178574

Simplifying
x = 2.569178574


Subproblem 2
x + 6.5 = -9.069178574

Simplifying
x + 6.5 = -9.069178574

Reorder the terms:
6.5 + x = -9.069178574

Solving
6.5 + x = -9.069178574

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 = -9.069178574 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + x = -9.069178574 + -6.5
x = -9.069178574 + -6.5

Combine like terms: -9.069178574 + -6.5 = -15.569178574
x = -15.569178574

Simplifying
x = -15.569178574


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.569178574, -15.569178574}

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