n^2+7n-72=0

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Solution for n^2+7n-72=0 equation:


Simplifying
n2 + 7n + -72 = 0

Reorder the terms:
-72 + 7n + n2 = 0

Solving
-72 + 7n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + 7n + 72 + n2 = 0 + 72

Reorder the terms:
-72 + 72 + 7n + n2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + 7n + n2 = 0 + 72
7n + n2 = 0 + 72

Combine like terms: 0 + 72 = 72
7n + n2 = 72

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 72 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 72 + 12.25

Combine like terms: 72 + 12.25 = 84.25
12.25 + 7n + n2 = 84.25

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(n + 3.5) equal to 9.178779875 and -9.178779875.

Subproblem 1

n + 3.5 = 9.178779875 Simplifying n + 3.5 = 9.178779875 Reorder the terms: 3.5 + n = 9.178779875 Solving 3.5 + n = 9.178779875 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 9.178779875 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 9.178779875 + -3.5 n = 9.178779875 + -3.5 Combine like terms: 9.178779875 + -3.5 = 5.678779875 n = 5.678779875 Simplifying n = 5.678779875

Subproblem 2

n + 3.5 = -9.178779875 Simplifying n + 3.5 = -9.178779875 Reorder the terms: 3.5 + n = -9.178779875 Solving 3.5 + n = -9.178779875 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -9.178779875 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -9.178779875 + -3.5 n = -9.178779875 + -3.5 Combine like terms: -9.178779875 + -3.5 = -12.678779875 n = -12.678779875 Simplifying n = -12.678779875

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.678779875, -12.678779875}

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