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}
