3n^2+8n-40=0

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Solution for 3n^2+8n-40=0 equation:


Simplifying
3n2 + 8n + -40 = 0

Reorder the terms:
-40 + 8n + 3n2 = 0

Solving
-40 + 8n + 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'.
-13.33333333 + 2.666666667n + n2 = 0

Move the constant term to the right:

Add '13.33333333' to each side of the equation.
-13.33333333 + 2.666666667n + 13.33333333 + n2 = 0 + 13.33333333

Reorder the terms:
-13.33333333 + 13.33333333 + 2.666666667n + n2 = 0 + 13.33333333

Combine like terms: -13.33333333 + 13.33333333 = 0.00000000
0.00000000 + 2.666666667n + n2 = 0 + 13.33333333
2.666666667n + n2 = 0 + 13.33333333

Combine like terms: 0 + 13.33333333 = 13.33333333
2.666666667n + n2 = 13.33333333

The n term is 2.666666667n.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667n + 1.777777780 + n2 = 13.33333333 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667n + n2 = 13.33333333 + 1.777777780

Combine like terms: 13.33333333 + 1.777777780 = 15.11111111
1.777777780 + 2.666666667n + n2 = 15.11111111

Factor a perfect square on the left side:
(n + 1.333333334)(n + 1.333333334) = 15.11111111

Calculate the square root of the right side: 3.887301263

Break this problem into two subproblems by setting 
(n + 1.333333334) equal to 3.887301263 and -3.887301263.

Subproblem 1

n + 1.333333334 = 3.887301263 Simplifying n + 1.333333334 = 3.887301263 Reorder the terms: 1.333333334 + n = 3.887301263 Solving 1.333333334 + n = 3.887301263 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = 3.887301263 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 3.887301263 + -1.333333334 n = 3.887301263 + -1.333333334 Combine like terms: 3.887301263 + -1.333333334 = 2.553967929 n = 2.553967929 Simplifying n = 2.553967929

Subproblem 2

n + 1.333333334 = -3.887301263 Simplifying n + 1.333333334 = -3.887301263 Reorder the terms: 1.333333334 + n = -3.887301263 Solving 1.333333334 + n = -3.887301263 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = -3.887301263 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -3.887301263 + -1.333333334 n = -3.887301263 + -1.333333334 Combine like terms: -3.887301263 + -1.333333334 = -5.220634597 n = -5.220634597 Simplifying n = -5.220634597

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.553967929, -5.220634597}

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