10n+n^2=72

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Solution for 10n+n^2=72 equation:


Simplifying
10n + n2 = 72

Solving
10n + n2 = 72

Solving for variable 'n'.

Reorder the terms:
-72 + 10n + n2 = 72 + -72

Combine like terms: 72 + -72 = 0
-72 + 10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + 10n + 72 + n2 = 0 + 72

Reorder the terms:
-72 + 72 + 10n + n2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + 10n + n2 = 0 + 72
10n + n2 = 0 + 72

Combine like terms: 0 + 72 = 72
10n + n2 = 72

The n term is 10n.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10n + 25 + n2 = 72 + 25

Reorder the terms:
25 + 10n + n2 = 72 + 25

Combine like terms: 72 + 25 = 97
25 + 10n + n2 = 97

Factor a perfect square on the left side:
(n + 5)(n + 5) = 97

Calculate the square root of the right side: 9.848857802

Break this problem into two subproblems by setting 
(n + 5) equal to 9.848857802 and -9.848857802.

Subproblem 1

n + 5 = 9.848857802 Simplifying n + 5 = 9.848857802 Reorder the terms: 5 + n = 9.848857802 Solving 5 + n = 9.848857802 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 9.848857802 + -5 Combine like terms: 5 + -5 = 0 0 + n = 9.848857802 + -5 n = 9.848857802 + -5 Combine like terms: 9.848857802 + -5 = 4.848857802 n = 4.848857802 Simplifying n = 4.848857802

Subproblem 2

n + 5 = -9.848857802 Simplifying n + 5 = -9.848857802 Reorder the terms: 5 + n = -9.848857802 Solving 5 + n = -9.848857802 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -9.848857802 + -5 Combine like terms: 5 + -5 = 0 0 + n = -9.848857802 + -5 n = -9.848857802 + -5 Combine like terms: -9.848857802 + -5 = -14.848857802 n = -14.848857802 Simplifying n = -14.848857802

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.848857802, -14.848857802}

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