(n-5)(n-6)=120

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Solution for (n-5)(n-6)=120 equation:


Simplifying
(n + -5)(n + -6) = 120

Reorder the terms:
(-5 + n)(n + -6) = 120

Reorder the terms:
(-5 + n)(-6 + n) = 120

Multiply (-5 + n) * (-6 + n)
(-5(-6 + n) + n(-6 + n)) = 120
((-6 * -5 + n * -5) + n(-6 + n)) = 120
((30 + -5n) + n(-6 + n)) = 120
(30 + -5n + (-6 * n + n * n)) = 120
(30 + -5n + (-6n + n2)) = 120

Combine like terms: -5n + -6n = -11n
(30 + -11n + n2) = 120

Solving
30 + -11n + n2 = 120

Solving for variable 'n'.

Reorder the terms:
30 + -120 + -11n + n2 = 120 + -120

Combine like terms: 30 + -120 = -90
-90 + -11n + n2 = 120 + -120

Combine like terms: 120 + -120 = 0
-90 + -11n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + -11n + 90 + n2 = 0 + 90

Reorder the terms:
-90 + 90 + -11n + n2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + -11n + n2 = 0 + 90
-11n + n2 = 0 + 90

Combine like terms: 0 + 90 = 90
-11n + n2 = 90

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 = 90 + 30.25

Reorder the terms:
30.25 + -11n + n2 = 90 + 30.25

Combine like terms: 90 + 30.25 = 120.25
30.25 + -11n + n2 = 120.25

Factor a perfect square on the left side:
(n + -5.5)(n + -5.5) = 120.25

Calculate the square root of the right side: 10.9658561

Break this problem into two subproblems by setting 
(n + -5.5) equal to 10.9658561 and -10.9658561.

Subproblem 1

n + -5.5 = 10.9658561 Simplifying n + -5.5 = 10.9658561 Reorder the terms: -5.5 + n = 10.9658561 Solving -5.5 + n = 10.9658561 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 = 10.9658561 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = 10.9658561 + 5.5 n = 10.9658561 + 5.5 Combine like terms: 10.9658561 + 5.5 = 16.4658561 n = 16.4658561 Simplifying n = 16.4658561

Subproblem 2

n + -5.5 = -10.9658561 Simplifying n + -5.5 = -10.9658561 Reorder the terms: -5.5 + n = -10.9658561 Solving -5.5 + n = -10.9658561 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 = -10.9658561 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = -10.9658561 + 5.5 n = -10.9658561 + 5.5 Combine like terms: -10.9658561 + 5.5 = -5.4658561 n = -5.4658561 Simplifying n = -5.4658561

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.4658561, -5.4658561}

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