2n(n+10)=75

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Solution for 2n(n+10)=75 equation:


Simplifying
2n(n + 10) = 75

Reorder the terms:
2n(10 + n) = 75
(10 * 2n + n * 2n) = 75
(20n + 2n2) = 75

Solving
20n + 2n2 = 75

Solving for variable 'n'.

Reorder the terms:
-75 + 20n + 2n2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 20n + 2n2 = 0

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

Divide each side by '2'.
-37.5 + 10n + n2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + 10n + 37.5 + n2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + 10n + n2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + 10n + n2 = 0 + 37.5
10n + n2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
10n + n2 = 37.5

The n term is 10n.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10n + 25 + n2 = 37.5 + 25

Reorder the terms:
25 + 10n + n2 = 37.5 + 25

Combine like terms: 37.5 + 25 = 62.5
25 + 10n + n2 = 62.5

Factor a perfect square on the left side:
(n + 5)(n + 5) = 62.5

Calculate the square root of the right side: 7.90569415

Break this problem into two subproblems by setting 
(n + 5) equal to 7.90569415 and -7.90569415.

Subproblem 1

n + 5 = 7.90569415 Simplifying n + 5 = 7.90569415 Reorder the terms: 5 + n = 7.90569415 Solving 5 + n = 7.90569415 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 7.90569415 + -5 Combine like terms: 5 + -5 = 0 0 + n = 7.90569415 + -5 n = 7.90569415 + -5 Combine like terms: 7.90569415 + -5 = 2.90569415 n = 2.90569415 Simplifying n = 2.90569415

Subproblem 2

n + 5 = -7.90569415 Simplifying n + 5 = -7.90569415 Reorder the terms: 5 + n = -7.90569415 Solving 5 + n = -7.90569415 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -7.90569415 + -5 Combine like terms: 5 + -5 = 0 0 + n = -7.90569415 + -5 n = -7.90569415 + -5 Combine like terms: -7.90569415 + -5 = -12.90569415 n = -12.90569415 Simplifying n = -12.90569415

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.90569415, -12.90569415}

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