(2n)(2n)+n-120=0

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Solution for (2n)(2n)+n-120=0 equation:


Simplifying
(2n)(2n) + n + -120 = 0

Remove parenthesis around (2n)
2n(2n) + n + -120 = 0

Remove parenthesis around (2n)
2n * 2n + n + -120 = 0

Reorder the terms for easier multiplication:
2 * 2n * n + n + -120 = 0

Multiply 2 * 2
4n * n + n + -120 = 0

Multiply n * n
4n2 + n + -120 = 0

Reorder the terms:
-120 + n + 4n2 = 0

Solving
-120 + n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-30 + 0.25n + n2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + 0.25n + 30 + n2 = 0 + 30

Reorder the terms:
-30 + 30 + 0.25n + n2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + 0.25n + n2 = 0 + 30
0.25n + n2 = 0 + 30

Combine like terms: 0 + 30 = 30
0.25n + n2 = 30

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.
0.25n + 0.25 + n2 = 30 + 0.25

Reorder the terms:
0.25 + 0.25n + n2 = 30 + 0.25

Combine like terms: 30 + 0.25 = 30.25
0.25 + 0.25n + n2 = 30.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 30.25

Calculate the square root of the right side: 5.5

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.5 and -5.5.

Subproblem 1

n + 0.5 = 5.5 Simplifying n + 0.5 = 5.5 Reorder the terms: 0.5 + n = 5.5 Solving 0.5 + n = 5.5 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 = 5.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 5.5 + -0.5 n = 5.5 + -0.5 Combine like terms: 5.5 + -0.5 = 5 n = 5 Simplifying n = 5

Subproblem 2

n + 0.5 = -5.5 Simplifying n + 0.5 = -5.5 Reorder the terms: 0.5 + n = -5.5 Solving 0.5 + n = -5.5 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 = -5.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -5.5 + -0.5 n = -5.5 + -0.5 Combine like terms: -5.5 + -0.5 = -6 n = -6 Simplifying n = -6

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5, -6}

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