3n^2-14n-15=0

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Solution for 3n^2-14n-15=0 equation: 


Simplifying
3n2 + -14n + -15 = 0

Reorder the terms:
-15 + -14n + 3n2 = 0

Solving
-15 + -14n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-5 + -4.666666667n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -4.666666667n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -4.666666667n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -4.666666667n + n2 = 0 + 5
-4.666666667n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-4.666666667n + n2 = 5

The n term is -4.666666667n.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667n + 5.444444448 + n2 = 5 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667n + n2 = 5 + 5.444444448

Combine like terms: 5 + 5.444444448 = 10.444444448
5.444444448 + -4.666666667n + n2 = 10.444444448

Factor a perfect square on the left side:
(n + -2.333333334)(n + -2.333333334) = 10.444444448

Calculate the square root of the right side: 3.231786572

Break this problem into two subproblems by setting 
(n + -2.333333334) equal to 3.231786572 and -3.231786572.


Subproblem 1
n + -2.333333334 = 3.231786572

Simplifying
n + -2.333333334 = 3.231786572

Reorder the terms:
-2.333333334 + n = 3.231786572

Solving
-2.333333334 + n = 3.231786572

Solving for variable 'n'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + n = 3.231786572 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + n = 3.231786572 + 2.333333334
n = 3.231786572 + 2.333333334

Combine like terms: 3.231786572 + 2.333333334 = 5.565119906
n = 5.565119906

Simplifying
n = 5.565119906


Subproblem 2
n + -2.333333334 = -3.231786572

Simplifying
n + -2.333333334 = -3.231786572

Reorder the terms:
-2.333333334 + n = -3.231786572

Solving
-2.333333334 + n = -3.231786572

Solving for variable 'n'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + n = -3.231786572 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + n = -3.231786572 + 2.333333334
n = -3.231786572 + 2.333333334

Combine like terms: -3.231786572 + 2.333333334 = -0.898453238
n = -0.898453238

Simplifying
n = -0.898453238


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.565119906, -0.898453238}

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