n^2+n=50

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Solution for n^2+n=50 equation:


Simplifying
n2 + n = 50

Reorder the terms:
n + n2 = 50

Solving
n + n2 = 50

Solving for variable 'n'.

Reorder the terms:
-50 + n + n2 = 50 + -50

Combine like terms: 50 + -50 = 0
-50 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + n + 50 + n2 = 0 + 50

Reorder the terms:
-50 + 50 + n + n2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + n + n2 = 0 + 50
n + n2 = 0 + 50

Combine like terms: 0 + 50 = 50
n + n2 = 50

The n term is n.  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.
n + 0.25 + n2 = 50 + 0.25

Reorder the terms:
0.25 + n + n2 = 50 + 0.25

Combine like terms: 50 + 0.25 = 50.25
0.25 + n + n2 = 50.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 50.25

Calculate the square root of the right side: 7.088723439

Break this problem into two subproblems by setting 
(n + 0.5) equal to 7.088723439 and -7.088723439.

Subproblem 1

n + 0.5 = 7.088723439 Simplifying n + 0.5 = 7.088723439 Reorder the terms: 0.5 + n = 7.088723439 Solving 0.5 + n = 7.088723439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 7.088723439 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 7.088723439 + -0.5 n = 7.088723439 + -0.5 Combine like terms: 7.088723439 + -0.5 = 6.588723439 n = 6.588723439 Simplifying n = 6.588723439

Subproblem 2

n + 0.5 = -7.088723439 Simplifying n + 0.5 = -7.088723439 Reorder the terms: 0.5 + n = -7.088723439 Solving 0.5 + n = -7.088723439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -7.088723439 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -7.088723439 + -0.5 n = -7.088723439 + -0.5 Combine like terms: -7.088723439 + -0.5 = -7.588723439 n = -7.588723439 Simplifying n = -7.588723439

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.588723439, -7.588723439}

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