(n^2*2)+(n*2)=50

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Solution for (n^2*2)+(n*2)=50 equation:


Simplifying
(n2 * 2) + (n * 2) = 50

Reorder the terms for easier multiplication:
(2n2) + (n * 2) = 50

Reorder the terms for easier multiplication:
(2n2) + (2n) = 50

Reorder the terms:
(2n) + (2n2) = 50

Solving
(2n) + (2n2) = 50

Solving for variable 'n'.

Reorder the terms:
-50 + (2n) + (2n2) = 50 + -50

Combine like terms: 50 + -50 = 0
-50 + (2n) + (2n2) = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-25 + n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-25 + n + n2)' equal to zero and attempt to solve:

Simplifying
-25 + n + n2 = 0

Solving
-25 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + n + 25 + n2 = 0 + 25

Reorder the terms:
-25 + 25 + n + n2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + n + n2 = 0 + 25
n + n2 = 0 + 25

Combine like terms: 0 + 25 = 25
n + n2 = 25

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 25 + 0.25

Reorder the terms:
0.25 + n + n2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + n + n2 = 25.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.024937811 and -5.024937811.


Subproblem 1
n + 0.5 = 5.024937811

Simplifying
n + 0.5 = 5.024937811

Reorder the terms:
0.5 + n = 5.024937811

Solving
0.5 + n = 5.024937811

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 5.024937811 + -0.5
n = 5.024937811 + -0.5

Combine like terms: 5.024937811 + -0.5 = 4.524937811
n = 4.524937811

Simplifying
n = 4.524937811


Subproblem 2
n + 0.5 = -5.024937811

Simplifying
n + 0.5 = -5.024937811

Reorder the terms:
0.5 + n = -5.024937811

Solving
0.5 + n = -5.024937811

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -5.024937811 + -0.5
n = -5.024937811 + -0.5

Combine like terms: -5.024937811 + -0.5 = -5.524937811
n = -5.524937811

Simplifying
n = -5.524937811


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.524937811, -5.524937811}
Solution
n = {4.524937811, -5.524937811}

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