x(x+1)=85

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Solution for x(x+1)=85 equation: 


Simplifying
x(x + 1) = 85

Reorder the terms:
x(1 + x) = 85
(1 * x + x * x) = 85
(1x + x2) = 85

Solving
1x + x2 = 85

Solving for variable 'x'.

Reorder the terms:
-85 + 1x + x2 = 85 + -85

Combine like terms: 85 + -85 = 0
-85 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '85' to each side of the equation.
-85 + 1x + 85 + x2 = 0 + 85

Reorder the terms:
-85 + 85 + 1x + x2 = 0 + 85

Combine like terms: -85 + 85 = 0
0 + 1x + x2 = 0 + 85
1x + x2 = 0 + 85

Combine like terms: 0 + 85 = 85
1x + x2 = 85

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 = 85 + 0.25

Reorder the terms:
0.25 + x + x2 = 85 + 0.25

Combine like terms: 85 + 0.25 = 85.25
0.25 + x + x2 = 85.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 85.25

Calculate the square root of the right side: 9.233092656

Break this problem into two subproblems by setting 
(x + 0.5) equal to 9.233092656 and -9.233092656.


Subproblem 1
x + 0.5 = 9.233092656

Simplifying
x + 0.5 = 9.233092656

Reorder the terms:
0.5 + x = 9.233092656

Solving
0.5 + x = 9.233092656

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 = 9.233092656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 9.233092656 + -0.5
x = 9.233092656 + -0.5

Combine like terms: 9.233092656 + -0.5 = 8.733092656
x = 8.733092656

Simplifying
x = 8.733092656


Subproblem 2
x + 0.5 = -9.233092656

Simplifying
x + 0.5 = -9.233092656

Reorder the terms:
0.5 + x = -9.233092656

Solving
0.5 + x = -9.233092656

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 = -9.233092656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -9.233092656 + -0.5
x = -9.233092656 + -0.5

Combine like terms: -9.233092656 + -0.5 = -9.733092656
x = -9.733092656

Simplifying
x = -9.733092656


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.733092656, -9.733092656}

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