X(X+1)=180

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Solution for X(X+1)=180 equation:


Simplifying
X(X + 1) = 180

Reorder the terms:
X(1 + X) = 180
(1 * X + X * X) = 180
(1X + X2) = 180

Solving
1X + X2 = 180

Solving for variable 'X'.

Reorder the terms:
-180 + 1X + X2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 1X + X2 = 0

Begin completing the square.

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + 1X + 180 + X2 = 0 + 180

Reorder the terms:
-180 + 180 + 1X + X2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + 1X + X2 = 0 + 180
1X + X2 = 0 + 180

Combine like terms: 0 + 180 = 180
1X + X2 = 180

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 = 180 + 0.25

Reorder the terms:
0.25 + X + X2 = 180 + 0.25

Combine like terms: 180 + 0.25 = 180.25
0.25 + X + X2 = 180.25

Factor a perfect square on the left side:
(X + 0.5)(X + 0.5) = 180.25

Calculate the square root of the right side: 13.425721582

Break this problem into two subproblems by setting 
(X + 0.5) equal to 13.425721582 and -13.425721582.

Subproblem 1

X + 0.5 = 13.425721582 Simplifying X + 0.5 = 13.425721582 Reorder the terms: 0.5 + X = 13.425721582 Solving 0.5 + X = 13.425721582 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 = 13.425721582 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 13.425721582 + -0.5 X = 13.425721582 + -0.5 Combine like terms: 13.425721582 + -0.5 = 12.925721582 X = 12.925721582 Simplifying X = 12.925721582

Subproblem 2

X + 0.5 = -13.425721582 Simplifying X + 0.5 = -13.425721582 Reorder the terms: 0.5 + X = -13.425721582 Solving 0.5 + X = -13.425721582 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 = -13.425721582 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -13.425721582 + -0.5 X = -13.425721582 + -0.5 Combine like terms: -13.425721582 + -0.5 = -13.925721582 X = -13.925721582 Simplifying X = -13.925721582

Solution

The solution to the problem is based on the solutions from the subproblems. X = {12.925721582, -13.925721582}

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