n^2+3n=60

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Solution for n^2+3n=60 equation:


Simplifying
n2 + 3n = 60

Reorder the terms:
3n + n2 = 60

Solving
3n + n2 = 60

Solving for variable 'n'.

Reorder the terms:
-60 + 3n + n2 = 60 + -60

Combine like terms: 60 + -60 = 0
-60 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 3n + 60 + n2 = 0 + 60

Reorder the terms:
-60 + 60 + 3n + n2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 3n + n2 = 0 + 60
3n + n2 = 0 + 60

Combine like terms: 0 + 60 = 60
3n + n2 = 60

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 60 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 60 + 2.25

Combine like terms: 60 + 2.25 = 62.25
2.25 + 3n + n2 = 62.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 62.25

Calculate the square root of the right side: 7.889866919

Break this problem into two subproblems by setting 
(n + 1.5) equal to 7.889866919 and -7.889866919.

Subproblem 1

n + 1.5 = 7.889866919 Simplifying n + 1.5 = 7.889866919 Reorder the terms: 1.5 + n = 7.889866919 Solving 1.5 + n = 7.889866919 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 7.889866919 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 7.889866919 + -1.5 n = 7.889866919 + -1.5 Combine like terms: 7.889866919 + -1.5 = 6.389866919 n = 6.389866919 Simplifying n = 6.389866919

Subproblem 2

n + 1.5 = -7.889866919 Simplifying n + 1.5 = -7.889866919 Reorder the terms: 1.5 + n = -7.889866919 Solving 1.5 + n = -7.889866919 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -7.889866919 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -7.889866919 + -1.5 n = -7.889866919 + -1.5 Combine like terms: -7.889866919 + -1.5 = -9.389866919 n = -9.389866919 Simplifying n = -9.389866919

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.389866919, -9.389866919}

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