n^2-23n=60

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Solution for n^2-23n=60 equation:


Simplifying
n2 + -23n = 60

Reorder the terms:
-23n + n2 = 60

Solving
-23n + n2 = 60

Solving for variable 'n'.

Reorder the terms:
-60 + -23n + n2 = 60 + -60

Combine like terms: 60 + -60 = 0
-60 + -23n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + -23n + 60 + n2 = 0 + 60

Reorder the terms:
-60 + 60 + -23n + n2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + -23n + n2 = 0 + 60
-23n + n2 = 0 + 60

Combine like terms: 0 + 60 = 60
-23n + n2 = 60

The n term is -23n.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23n + 132.25 + n2 = 60 + 132.25

Reorder the terms:
132.25 + -23n + n2 = 60 + 132.25

Combine like terms: 60 + 132.25 = 192.25
132.25 + -23n + n2 = 192.25

Factor a perfect square on the left side:
(n + -11.5)(n + -11.5) = 192.25

Calculate the square root of the right side: 13.865424624

Break this problem into two subproblems by setting 
(n + -11.5) equal to 13.865424624 and -13.865424624.

Subproblem 1

n + -11.5 = 13.865424624 Simplifying n + -11.5 = 13.865424624 Reorder the terms: -11.5 + n = 13.865424624 Solving -11.5 + n = 13.865424624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + n = 13.865424624 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + n = 13.865424624 + 11.5 n = 13.865424624 + 11.5 Combine like terms: 13.865424624 + 11.5 = 25.365424624 n = 25.365424624 Simplifying n = 25.365424624

Subproblem 2

n + -11.5 = -13.865424624 Simplifying n + -11.5 = -13.865424624 Reorder the terms: -11.5 + n = -13.865424624 Solving -11.5 + n = -13.865424624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + n = -13.865424624 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + n = -13.865424624 + 11.5 n = -13.865424624 + 11.5 Combine like terms: -13.865424624 + 11.5 = -2.365424624 n = -2.365424624 Simplifying n = -2.365424624

Solution

The solution to the problem is based on the solutions from the subproblems. n = {25.365424624, -2.365424624}

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