3n(n)+60n+46=0

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Solution for 3n(n)+60n+46=0 equation:


Simplifying
3n(n) + 60n + 46 = 0

Multiply n * n
3n2 + 60n + 46 = 0

Reorder the terms:
46 + 60n + 3n2 = 0

Solving
46 + 60n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
15.33333333 + 20n + n2 = 0

Move the constant term to the right:

Add '-15.33333333' to each side of the equation.
15.33333333 + 20n + -15.33333333 + n2 = 0 + -15.33333333

Reorder the terms:
15.33333333 + -15.33333333 + 20n + n2 = 0 + -15.33333333

Combine like terms: 15.33333333 + -15.33333333 = 0.00000000
0.00000000 + 20n + n2 = 0 + -15.33333333
20n + n2 = 0 + -15.33333333

Combine like terms: 0 + -15.33333333 = -15.33333333
20n + n2 = -15.33333333

The n term is 20n.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20n + 100 + n2 = -15.33333333 + 100

Reorder the terms:
100 + 20n + n2 = -15.33333333 + 100

Combine like terms: -15.33333333 + 100 = 84.66666667
100 + 20n + n2 = 84.66666667

Factor a perfect square on the left side:
(n + 10)(n + 10) = 84.66666667

Calculate the square root of the right side: 9.201449161

Break this problem into two subproblems by setting 
(n + 10) equal to 9.201449161 and -9.201449161.

Subproblem 1

n + 10 = 9.201449161 Simplifying n + 10 = 9.201449161 Reorder the terms: 10 + n = 9.201449161 Solving 10 + n = 9.201449161 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + n = 9.201449161 + -10 Combine like terms: 10 + -10 = 0 0 + n = 9.201449161 + -10 n = 9.201449161 + -10 Combine like terms: 9.201449161 + -10 = -0.798550839 n = -0.798550839 Simplifying n = -0.798550839

Subproblem 2

n + 10 = -9.201449161 Simplifying n + 10 = -9.201449161 Reorder the terms: 10 + n = -9.201449161 Solving 10 + n = -9.201449161 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + n = -9.201449161 + -10 Combine like terms: 10 + -10 = 0 0 + n = -9.201449161 + -10 n = -9.201449161 + -10 Combine like terms: -9.201449161 + -10 = -19.201449161 n = -19.201449161 Simplifying n = -19.201449161

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.798550839, -19.201449161}

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