6n^2-21n-15=0

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Solution for 6n^2-21n-15=0 equation: 


Simplifying
6n2 + -21n + -15 = 0

Reorder the terms:
-15 + -21n + 6n2 = 0

Solving
-15 + -21n + 6n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-5 + -7n + 2n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-5 + -7n + 2n2)' equal to zero and attempt to solve:

Simplifying
-5 + -7n + 2n2 = 0

Solving
-5 + -7n + 2n2 = 0

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

Divide each side by '2'.
-2.5 + -3.5n + n2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -3.5n + 2.5 + n2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -3.5n + n2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -3.5n + n2 = 0 + 2.5
-3.5n + n2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-3.5n + n2 = 2.5

The n term is -3.5n.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5n + 3.0625 + n2 = 2.5 + 3.0625

Reorder the terms:
3.0625 + -3.5n + n2 = 2.5 + 3.0625

Combine like terms: 2.5 + 3.0625 = 5.5625
3.0625 + -3.5n + n2 = 5.5625

Factor a perfect square on the left side:
(n + -1.75)(n + -1.75) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(n + -1.75) equal to 2.358495283 and -2.358495283.


Subproblem 1
n + -1.75 = 2.358495283

Simplifying
n + -1.75 = 2.358495283

Reorder the terms:
-1.75 + n = 2.358495283

Solving
-1.75 + n = 2.358495283

Solving for variable 'n'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + n = 2.358495283 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + n = 2.358495283 + 1.75
n = 2.358495283 + 1.75

Combine like terms: 2.358495283 + 1.75 = 4.108495283
n = 4.108495283

Simplifying
n = 4.108495283


Subproblem 2
n + -1.75 = -2.358495283

Simplifying
n + -1.75 = -2.358495283

Reorder the terms:
-1.75 + n = -2.358495283

Solving
-1.75 + n = -2.358495283

Solving for variable 'n'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + n = -2.358495283 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + n = -2.358495283 + 1.75
n = -2.358495283 + 1.75

Combine like terms: -2.358495283 + 1.75 = -0.608495283
n = -0.608495283

Simplifying
n = -0.608495283


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.108495283, -0.608495283}
Solution
n = {4.108495283, -0.608495283}

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