750+22n(n-12)=650+30(n-10)

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Solution for 750+22n(n-12)=650+30(n-10) equation: 


Simplifying
750 + 22n(n + -12) = 650 + 30(n + -10)

Reorder the terms:
750 + 22n(-12 + n) = 650 + 30(n + -10)
750 + (-12 * 22n + n * 22n) = 650 + 30(n + -10)
750 + (-264n + 22n2) = 650 + 30(n + -10)

Reorder the terms:
750 + -264n + 22n2 = 650 + 30(-10 + n)
750 + -264n + 22n2 = 650 + (-10 * 30 + n * 30)
750 + -264n + 22n2 = 650 + (-300 + 30n)

Combine like terms: 650 + -300 = 350
750 + -264n + 22n2 = 350 + 30n

Solving
750 + -264n + 22n2 = 350 + 30n

Solving for variable 'n'.

Reorder the terms:
750 + -350 + -264n + -30n + 22n2 = 350 + 30n + -350 + -30n

Combine like terms: 750 + -350 = 400
400 + -264n + -30n + 22n2 = 350 + 30n + -350 + -30n

Combine like terms: -264n + -30n = -294n
400 + -294n + 22n2 = 350 + 30n + -350 + -30n

Reorder the terms:
400 + -294n + 22n2 = 350 + -350 + 30n + -30n

Combine like terms: 350 + -350 = 0
400 + -294n + 22n2 = 0 + 30n + -30n
400 + -294n + 22n2 = 30n + -30n

Combine like terms: 30n + -30n = 0
400 + -294n + 22n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(200 + -147n + 11n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(200 + -147n + 11n2)' equal to zero and attempt to solve:

Simplifying
200 + -147n + 11n2 = 0

Solving
200 + -147n + 11n2 = 0

Begin completing the square.  Divide all terms by
11 the coefficient of the squared term: 

Divide each side by '11'.
18.18181818 + -13.36363636n + n2 = 0

Move the constant term to the right:

Add '-18.18181818' to each side of the equation.
18.18181818 + -13.36363636n + -18.18181818 + n2 = 0 + -18.18181818

Reorder the terms:
18.18181818 + -18.18181818 + -13.36363636n + n2 = 0 + -18.18181818

Combine like terms: 18.18181818 + -18.18181818 = 0.00000000
0.00000000 + -13.36363636n + n2 = 0 + -18.18181818
-13.36363636n + n2 = 0 + -18.18181818

Combine like terms: 0 + -18.18181818 = -18.18181818
-13.36363636n + n2 = -18.18181818

The n term is -13.36363636n.  Take half its coefficient (-6.68181818).
Square it (44.64669419) and add it to both sides.

Add '44.64669419' to each side of the equation.
-13.36363636n + 44.64669419 + n2 = -18.18181818 + 44.64669419

Reorder the terms:
44.64669419 + -13.36363636n + n2 = -18.18181818 + 44.64669419

Combine like terms: -18.18181818 + 44.64669419 = 26.46487601
44.64669419 + -13.36363636n + n2 = 26.46487601

Factor a perfect square on the left side:
(n + -6.68181818)(n + -6.68181818) = 26.46487601

Calculate the square root of the right side: 5.144402396

Break this problem into two subproblems by setting 
(n + -6.68181818) equal to 5.144402396 and -5.144402396.


Subproblem 1
n + -6.68181818 = 5.144402396

Simplifying
n + -6.68181818 = 5.144402396

Reorder the terms:
-6.68181818 + n = 5.144402396

Solving
-6.68181818 + n = 5.144402396

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '6.68181818' to each side of the equation.
-6.68181818 + 6.68181818 + n = 5.144402396 + 6.68181818

Combine like terms: -6.68181818 + 6.68181818 = 0.00000000
0.00000000 + n = 5.144402396 + 6.68181818
n = 5.144402396 + 6.68181818

Combine like terms: 5.144402396 + 6.68181818 = 11.826220576
n = 11.826220576

Simplifying
n = 11.826220576


Subproblem 2
n + -6.68181818 = -5.144402396

Simplifying
n + -6.68181818 = -5.144402396

Reorder the terms:
-6.68181818 + n = -5.144402396

Solving
-6.68181818 + n = -5.144402396

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '6.68181818' to each side of the equation.
-6.68181818 + 6.68181818 + n = -5.144402396 + 6.68181818

Combine like terms: -6.68181818 + 6.68181818 = 0.00000000
0.00000000 + n = -5.144402396 + 6.68181818
n = -5.144402396 + 6.68181818

Combine like terms: -5.144402396 + 6.68181818 = 1.537415784
n = 1.537415784

Simplifying
n = 1.537415784


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {11.826220576, 1.537415784}
Solution
n = {11.826220576, 1.537415784}

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