3n^2+n-7=0

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Solution for 3n^2+n-7=0 equation:


Simplifying
3n2 + n + -7 = 0

Reorder the terms:
-7 + n + 3n2 = 0

Solving
-7 + n + 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'.
-2.333333333 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + 0.3333333333n + 2.333333333 + n2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + 0.3333333333n + n2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + 0.3333333333n + n2 = 0 + 2.333333333
0.3333333333n + n2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
0.3333333333n + n2 = 2.333333333

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.
0.3333333333n + 0.25 + n2 = 2.333333333 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 2.333333333 + 0.25

Combine like terms: 2.333333333 + 0.25 = 2.583333333
0.25 + 0.3333333333n + n2 = 2.583333333

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 2.583333333

Calculate the square root of the right side: 1.607275127

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.607275127 and -1.607275127.

Subproblem 1

n + 0.5 = 1.607275127 Simplifying n + 0.5 = 1.607275127 Reorder the terms: 0.5 + n = 1.607275127 Solving 0.5 + n = 1.607275127 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 = 1.607275127 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.607275127 + -0.5 n = 1.607275127 + -0.5 Combine like terms: 1.607275127 + -0.5 = 1.107275127 n = 1.107275127 Simplifying n = 1.107275127

Subproblem 2

n + 0.5 = -1.607275127 Simplifying n + 0.5 = -1.607275127 Reorder the terms: 0.5 + n = -1.607275127 Solving 0.5 + n = -1.607275127 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 = -1.607275127 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.607275127 + -0.5 n = -1.607275127 + -0.5 Combine like terms: -1.607275127 + -0.5 = -2.107275127 n = -2.107275127 Simplifying n = -2.107275127

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.107275127, -2.107275127}

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