2(2n^2-9n-20)=0

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Solution for 2(2n^2-9n-20)=0 equation: 


Simplifying
2(2n2 + -9n + -20) = 0

Reorder the terms:
2(-20 + -9n + 2n2) = 0
(-20 * 2 + -9n * 2 + 2n2 * 2) = 0
(-40 + -18n + 4n2) = 0

Solving
-40 + -18n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-20 + -9n + 2n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-20 + -9n + 2n2)' equal to zero and attempt to solve:

Simplifying
-20 + -9n + 2n2 = 0

Solving
-20 + -9n + 2n2 = 0

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

Divide each side by '2'.
-10 + -4.5n + n2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -4.5n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + -4.5n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -4.5n + n2 = 0 + 10
-4.5n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
-4.5n + n2 = 10

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 = 10 + 5.0625

Reorder the terms:
5.0625 + -4.5n + n2 = 10 + 5.0625

Combine like terms: 10 + 5.0625 = 15.0625
5.0625 + -4.5n + n2 = 15.0625

Factor a perfect square on the left side:
(n + -2.25)(n + -2.25) = 15.0625

Calculate the square root of the right side: 3.881043674

Break this problem into two subproblems by setting 
(n + -2.25) equal to 3.881043674 and -3.881043674.


Subproblem 1
n + -2.25 = 3.881043674

Simplifying
n + -2.25 = 3.881043674

Reorder the terms:
-2.25 + n = 3.881043674

Solving
-2.25 + n = 3.881043674

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 = 3.881043674 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = 3.881043674 + 2.25
n = 3.881043674 + 2.25

Combine like terms: 3.881043674 + 2.25 = 6.131043674
n = 6.131043674

Simplifying
n = 6.131043674


Subproblem 2
n + -2.25 = -3.881043674

Simplifying
n + -2.25 = -3.881043674

Reorder the terms:
-2.25 + n = -3.881043674

Solving
-2.25 + n = -3.881043674

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 = -3.881043674 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = -3.881043674 + 2.25
n = -3.881043674 + 2.25

Combine like terms: -3.881043674 + 2.25 = -1.631043674
n = -1.631043674

Simplifying
n = -1.631043674


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.131043674, -1.631043674}
Solution
n = {6.131043674, -1.631043674}

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