n^2+n-16=0

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Solution for n^2+n-16=0 equation:


Simplifying
n2 + n + -16 = 0

Reorder the terms:
-16 + n + n2 = 0

Solving
-16 + n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + n + 16 + n2 = 0 + 16

Reorder the terms:
-16 + 16 + n + n2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + n + n2 = 0 + 16
n + n2 = 0 + 16

Combine like terms: 0 + 16 = 16
n + n2 = 16

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 = 16 + 0.25

Reorder the terms:
0.25 + n + n2 = 16 + 0.25

Combine like terms: 16 + 0.25 = 16.25
0.25 + n + n2 = 16.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(n + 0.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

n + 0.5 = 4.031128874 Simplifying n + 0.5 = 4.031128874 Reorder the terms: 0.5 + n = 4.031128874 Solving 0.5 + n = 4.031128874 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 = 4.031128874 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 4.031128874 + -0.5 n = 4.031128874 + -0.5 Combine like terms: 4.031128874 + -0.5 = 3.531128874 n = 3.531128874 Simplifying n = 3.531128874

Subproblem 2

n + 0.5 = -4.031128874 Simplifying n + 0.5 = -4.031128874 Reorder the terms: 0.5 + n = -4.031128874 Solving 0.5 + n = -4.031128874 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 = -4.031128874 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -4.031128874 + -0.5 n = -4.031128874 + -0.5 Combine like terms: -4.031128874 + -0.5 = -4.531128874 n = -4.531128874 Simplifying n = -4.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.531128874, -4.531128874}

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