10n^2-65n-345=0

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Solution for 10n^2-65n-345=0 equation: 


Simplifying
10n2 + -65n + -345 = 0

Reorder the terms:
-345 + -65n + 10n2 = 0

Solving
-345 + -65n + 10n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '5'.
5(-69 + -13n + 2n2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-69 + -13n + 2n2)' equal to zero and attempt to solve:

Simplifying
-69 + -13n + 2n2 = 0

Solving
-69 + -13n + 2n2 = 0

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

Divide each side by '2'.
-34.5 + -6.5n + n2 = 0

Move the constant term to the right:

Add '34.5' to each side of the equation.
-34.5 + -6.5n + 34.5 + n2 = 0 + 34.5

Reorder the terms:
-34.5 + 34.5 + -6.5n + n2 = 0 + 34.5

Combine like terms: -34.5 + 34.5 = 0.0
0.0 + -6.5n + n2 = 0 + 34.5
-6.5n + n2 = 0 + 34.5

Combine like terms: 0 + 34.5 = 34.5
-6.5n + n2 = 34.5

The n term is -6.5n.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5n + 10.5625 + n2 = 34.5 + 10.5625

Reorder the terms:
10.5625 + -6.5n + n2 = 34.5 + 10.5625

Combine like terms: 34.5 + 10.5625 = 45.0625
10.5625 + -6.5n + n2 = 45.0625

Factor a perfect square on the left side:
(n + -3.25)(n + -3.25) = 45.0625

Calculate the square root of the right side: 6.712860791

Break this problem into two subproblems by setting 
(n + -3.25) equal to 6.712860791 and -6.712860791.


Subproblem 1
n + -3.25 = 6.712860791

Simplifying
n + -3.25 = 6.712860791

Reorder the terms:
-3.25 + n = 6.712860791

Solving
-3.25 + n = 6.712860791

Solving for variable 'n'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + n = 6.712860791 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + n = 6.712860791 + 3.25
n = 6.712860791 + 3.25

Combine like terms: 6.712860791 + 3.25 = 9.962860791
n = 9.962860791

Simplifying
n = 9.962860791


Subproblem 2
n + -3.25 = -6.712860791

Simplifying
n + -3.25 = -6.712860791

Reorder the terms:
-3.25 + n = -6.712860791

Solving
-3.25 + n = -6.712860791

Solving for variable 'n'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + n = -6.712860791 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + n = -6.712860791 + 3.25
n = -6.712860791 + 3.25

Combine like terms: -6.712860791 + 3.25 = -3.462860791
n = -3.462860791

Simplifying
n = -3.462860791


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9.962860791, -3.462860791}
Solution
n = {9.962860791, -3.462860791}

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