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Simplifying
2n2 + -22n + 60 = 0

Reorder the terms:
60 + -22n + 2n2 = 0

Solving
60 + -22n + 2n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(30 + -11n + n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(30 + -11n + n2)' equal to zero and attempt to solve: Simplifying 30 + -11n + n2 = 0 Solving 30 + -11n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-30' to each side of the equation. 30 + -11n + -30 + n2 = 0 + -30 Reorder the terms: 30 + -30 + -11n + n2 = 0 + -30 Combine like terms: 30 + -30 = 0 0 + -11n + n2 = 0 + -30 -11n + n2 = 0 + -30 Combine like terms: 0 + -30 = -30 -11n + n2 = -30 The n term is -11n. Take half its coefficient (-5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. -11n + 30.25 + n2 = -30 + 30.25 Reorder the terms: 30.25 + -11n + n2 = -30 + 30.25 Combine like terms: -30 + 30.25 = 0.25 30.25 + -11n + n2 = 0.25 Factor a perfect square on the left side: (n + -5.5)(n + -5.5) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (n + -5.5) equal to 0.5 and -0.5.

Subproblem 1

n + -5.5 = 0.5 Simplifying n + -5.5 = 0.5 Reorder the terms: -5.5 + n = 0.5 Solving -5.5 + n = 0.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + n = 0.5 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = 0.5 + 5.5 n = 0.5 + 5.5 Combine like terms: 0.5 + 5.5 = 6 n = 6 Simplifying n = 6

Subproblem 2

n + -5.5 = -0.5 Simplifying n + -5.5 = -0.5 Reorder the terms: -5.5 + n = -0.5 Solving -5.5 + n = -0.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + n = -0.5 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = -0.5 + 5.5 n = -0.5 + 5.5 Combine like terms: -0.5 + 5.5 = 5 n = 5 Simplifying n = 5

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6, 5}

Solution

n = {6, 5}
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