x(x+1)=39

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Solution for x(x+1)=39 equation: 


Simplifying
x(x + 1) = 39

Reorder the terms:
x(1 + x) = 39
(1 * x + x * x) = 39
(1x + x2) = 39

Solving
1x + x2 = 39

Solving for variable 'x'.

Reorder the terms:
-39 + 1x + x2 = 39 + -39

Combine like terms: 39 + -39 = 0
-39 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 1x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + 1x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 1x + x2 = 0 + 39
1x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
1x + x2 = 39

The x term is 1x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 39 + 0.25

Reorder the terms:
0.25 + x + x2 = 39 + 0.25

Combine like terms: 39 + 0.25 = 39.25
0.25 + x + x2 = 39.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 39.25

Calculate the square root of the right side: 6.264982043

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.264982043 and -6.264982043.


Subproblem 1
x + 0.5 = 6.264982043

Simplifying
x + 0.5 = 6.264982043

Reorder the terms:
0.5 + x = 6.264982043

Solving
0.5 + x = 6.264982043

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 6.264982043 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 6.264982043 + -0.5
x = 6.264982043 + -0.5

Combine like terms: 6.264982043 + -0.5 = 5.764982043
x = 5.764982043

Simplifying
x = 5.764982043


Subproblem 2
x + 0.5 = -6.264982043

Simplifying
x + 0.5 = -6.264982043

Reorder the terms:
0.5 + x = -6.264982043

Solving
0.5 + x = -6.264982043

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -6.264982043 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -6.264982043 + -0.5
x = -6.264982043 + -0.5

Combine like terms: -6.264982043 + -0.5 = -6.764982043
x = -6.764982043

Simplifying
x = -6.764982043


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.764982043, -6.764982043}

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