(3n-2)(3n+2)-(2n-3)=3n(n+7)-17

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Solution for (3n-2)(3n+2)-(2n-3)=3n(n+7)-17 equation: 


Simplifying
(3n + -2)(3n + 2) + -1(2n + -3) = 3n(n + 7) + -17

Reorder the terms:
(-2 + 3n)(3n + 2) + -1(2n + -3) = 3n(n + 7) + -17

Reorder the terms:
(-2 + 3n)(2 + 3n) + -1(2n + -3) = 3n(n + 7) + -17

Multiply (-2 + 3n) * (2 + 3n)
(-2(2 + 3n) + 3n * (2 + 3n)) + -1(2n + -3) = 3n(n + 7) + -17
((2 * -2 + 3n * -2) + 3n * (2 + 3n)) + -1(2n + -3) = 3n(n + 7) + -17
((-4 + -6n) + 3n * (2 + 3n)) + -1(2n + -3) = 3n(n + 7) + -17
(-4 + -6n + (2 * 3n + 3n * 3n)) + -1(2n + -3) = 3n(n + 7) + -17
(-4 + -6n + (6n + 9n2)) + -1(2n + -3) = 3n(n + 7) + -17

Combine like terms: -6n + 6n = 0
(-4 + 0 + 9n2) + -1(2n + -3) = 3n(n + 7) + -17
(-4 + 9n2) + -1(2n + -3) = 3n(n + 7) + -17

Reorder the terms:
-4 + 9n2 + -1(-3 + 2n) = 3n(n + 7) + -17
-4 + 9n2 + (-3 * -1 + 2n * -1) = 3n(n + 7) + -17
-4 + 9n2 + (3 + -2n) = 3n(n + 7) + -17

Reorder the terms:
-4 + 3 + -2n + 9n2 = 3n(n + 7) + -17

Combine like terms: -4 + 3 = -1
-1 + -2n + 9n2 = 3n(n + 7) + -17

Reorder the terms:
-1 + -2n + 9n2 = 3n(7 + n) + -17
-1 + -2n + 9n2 = (7 * 3n + n * 3n) + -17
-1 + -2n + 9n2 = (21n + 3n2) + -17

Reorder the terms:
-1 + -2n + 9n2 = -17 + 21n + 3n2

Solving
-1 + -2n + 9n2 = -17 + 21n + 3n2

Solving for variable 'n'.

Reorder the terms:
-1 + 17 + -2n + -21n + 9n2 + -3n2 = -17 + 21n + 3n2 + 17 + -21n + -3n2

Combine like terms: -1 + 17 = 16
16 + -2n + -21n + 9n2 + -3n2 = -17 + 21n + 3n2 + 17 + -21n + -3n2

Combine like terms: -2n + -21n = -23n
16 + -23n + 9n2 + -3n2 = -17 + 21n + 3n2 + 17 + -21n + -3n2

Combine like terms: 9n2 + -3n2 = 6n2
16 + -23n + 6n2 = -17 + 21n + 3n2 + 17 + -21n + -3n2

Reorder the terms:
16 + -23n + 6n2 = -17 + 17 + 21n + -21n + 3n2 + -3n2

Combine like terms: -17 + 17 = 0
16 + -23n + 6n2 = 0 + 21n + -21n + 3n2 + -3n2
16 + -23n + 6n2 = 21n + -21n + 3n2 + -3n2

Combine like terms: 21n + -21n = 0
16 + -23n + 6n2 = 0 + 3n2 + -3n2
16 + -23n + 6n2 = 3n2 + -3n2

Combine like terms: 3n2 + -3n2 = 0
16 + -23n + 6n2 = 0

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

Divide each side by '6'.
2.666666667 + -3.833333333n + n2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + -3.833333333n + -2.666666667 + n2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + -3.833333333n + n2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + -3.833333333n + n2 = 0 + -2.666666667
-3.833333333n + n2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
-3.833333333n + n2 = -2.666666667

The n term is -3.833333333n.  Take half its coefficient (-1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
-3.833333333n + 3.673611112 + n2 = -2.666666667 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333n + n2 = -2.666666667 + 3.673611112

Combine like terms: -2.666666667 + 3.673611112 = 1.006944445
3.673611112 + -3.833333333n + n2 = 1.006944445

Factor a perfect square on the left side:
(n + -1.916666667)(n + -1.916666667) = 1.006944445

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(n + -1.916666667) equal to 1.003466215 and -1.003466215.


Subproblem 1
n + -1.916666667 = 1.003466215

Simplifying
n + -1.916666667 = 1.003466215

Reorder the terms:
-1.916666667 + n = 1.003466215

Solving
-1.916666667 + n = 1.003466215

Solving for variable 'n'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + n = 1.003466215 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + n = 1.003466215 + 1.916666667
n = 1.003466215 + 1.916666667

Combine like terms: 1.003466215 + 1.916666667 = 2.920132882
n = 2.920132882

Simplifying
n = 2.920132882


Subproblem 2
n + -1.916666667 = -1.003466215

Simplifying
n + -1.916666667 = -1.003466215

Reorder the terms:
-1.916666667 + n = -1.003466215

Solving
-1.916666667 + n = -1.003466215

Solving for variable 'n'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + n = -1.003466215 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + n = -1.003466215 + 1.916666667
n = -1.003466215 + 1.916666667

Combine like terms: -1.003466215 + 1.916666667 = 0.913200452
n = 0.913200452

Simplifying
n = 0.913200452


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.920132882, 0.913200452}

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