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

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


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

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

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

Solving
750 + -264n + 22n2 = 650 + -300n + 30n2

Solving for variable 'n'.

Reorder the terms:
750 + -650 + -264n + 300n + 22n2 + -30n2 = 650 + -300n + 30n2 + -650 + 300n + -30n2

Combine like terms: 750 + -650 = 100
100 + -264n + 300n + 22n2 + -30n2 = 650 + -300n + 30n2 + -650 + 300n + -30n2

Combine like terms: -264n + 300n = 36n
100 + 36n + 22n2 + -30n2 = 650 + -300n + 30n2 + -650 + 300n + -30n2

Combine like terms: 22n2 + -30n2 = -8n2
100 + 36n + -8n2 = 650 + -300n + 30n2 + -650 + 300n + -30n2

Reorder the terms:
100 + 36n + -8n2 = 650 + -650 + -300n + 300n + 30n2 + -30n2

Combine like terms: 650 + -650 = 0
100 + 36n + -8n2 = 0 + -300n + 300n + 30n2 + -30n2
100 + 36n + -8n2 = -300n + 300n + 30n2 + -30n2

Combine like terms: -300n + 300n = 0
100 + 36n + -8n2 = 0 + 30n2 + -30n2
100 + 36n + -8n2 = 30n2 + -30n2

Combine like terms: 30n2 + -30n2 = 0
100 + 36n + -8n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(25 + 9n + -2n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(25 + 9n + -2n2)' equal to zero and attempt to solve:

Simplifying
25 + 9n + -2n2 = 0

Solving
25 + 9n + -2n2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-12.5 + -4.5n + n2 = 0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + -4.5n + 12.5 + n2 = 0 + 12.5

Reorder the terms:
-12.5 + 12.5 + -4.5n + n2 = 0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + -4.5n + n2 = 0 + 12.5
-4.5n + n2 = 0 + 12.5

Combine like terms: 0 + 12.5 = 12.5
-4.5n + n2 = 12.5

The n term is -4.5n.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5n + 5.0625 + n2 = 12.5 + 5.0625

Reorder the terms:
5.0625 + -4.5n + n2 = 12.5 + 5.0625

Combine like terms: 12.5 + 5.0625 = 17.5625
5.0625 + -4.5n + n2 = 17.5625

Factor a perfect square on the left side:
(n + -2.25)(n + -2.25) = 17.5625

Calculate the square root of the right side: 4.190763654

Break this problem into two subproblems by setting 
(n + -2.25) equal to 4.190763654 and -4.190763654.


Subproblem 1
n + -2.25 = 4.190763654

Simplifying
n + -2.25 = 4.190763654

Reorder the terms:
-2.25 + n = 4.190763654

Solving
-2.25 + n = 4.190763654

Solving for variable 'n'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + n = 4.190763654 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = 4.190763654 + 2.25
n = 4.190763654 + 2.25

Combine like terms: 4.190763654 + 2.25 = 6.440763654
n = 6.440763654

Simplifying
n = 6.440763654


Subproblem 2
n + -2.25 = -4.190763654

Simplifying
n + -2.25 = -4.190763654

Reorder the terms:
-2.25 + n = -4.190763654

Solving
-2.25 + n = -4.190763654

Solving for variable 'n'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + n = -4.190763654 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = -4.190763654 + 2.25
n = -4.190763654 + 2.25

Combine like terms: -4.190763654 + 2.25 = -1.940763654
n = -1.940763654

Simplifying
n = -1.940763654


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.440763654, -1.940763654}
Solution
n = {6.440763654, -1.940763654}

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