3n^2+6n-16=0

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Solution for 3n^2+6n-16=0 equation:


Simplifying
3n2 + 6n + -16 = 0

Reorder the terms:
-16 + 6n + 3n2 = 0

Solving
-16 + 6n + 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'.
-5.333333333 + 2n + n2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + 2n + 5.333333333 + n2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + 2n + n2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + 2n + n2 = 0 + 5.333333333
2n + n2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
2n + n2 = 5.333333333

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 5.333333333 + 1

Reorder the terms:
1 + 2n + n2 = 5.333333333 + 1

Combine like terms: 5.333333333 + 1 = 6.333333333
1 + 2n + n2 = 6.333333333

Factor a perfect square on the left side:
(n + 1)(n + 1) = 6.333333333

Calculate the square root of the right side: 2.516611478

Break this problem into two subproblems by setting 
(n + 1) equal to 2.516611478 and -2.516611478.

Subproblem 1

n + 1 = 2.516611478 Simplifying n + 1 = 2.516611478 Reorder the terms: 1 + n = 2.516611478 Solving 1 + n = 2.516611478 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 2.516611478 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.516611478 + -1 n = 2.516611478 + -1 Combine like terms: 2.516611478 + -1 = 1.516611478 n = 1.516611478 Simplifying n = 1.516611478

Subproblem 2

n + 1 = -2.516611478 Simplifying n + 1 = -2.516611478 Reorder the terms: 1 + n = -2.516611478 Solving 1 + n = -2.516611478 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -2.516611478 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.516611478 + -1 n = -2.516611478 + -1 Combine like terms: -2.516611478 + -1 = -3.516611478 n = -3.516611478 Simplifying n = -3.516611478

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.516611478, -3.516611478}

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