3n^2-2n-6=9

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Solution for 3n^2-2n-6=9 equation:


Simplifying
3n2 + -2n + -6 = 9

Reorder the terms:
-6 + -2n + 3n2 = 9

Solving
-6 + -2n + 3n2 = 9

Solving for variable 'n'.

Reorder the terms:
-6 + -9 + -2n + 3n2 = 9 + -9

Combine like terms: -6 + -9 = -15
-15 + -2n + 3n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-15 + -2n + 3n2 = 0

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

Divide each side by '3'.
-5 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.6666666667n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.6666666667n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.6666666667n + n2 = 0 + 5
-0.6666666667n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.6666666667n + n2 = 5

The n term is -0.6666666667n.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667n + 0.1111111112 + n2 = 5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 5 + 0.1111111112

Combine like terms: 5 + 0.1111111112 = 5.1111111112
0.1111111112 + -0.6666666667n + n2 = 5.1111111112

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 5.1111111112

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 2.260776661 and -2.260776661.

Subproblem 1

n + -0.3333333334 = 2.260776661 Simplifying n + -0.3333333334 = 2.260776661 Reorder the terms: -0.3333333334 + n = 2.260776661 Solving -0.3333333334 + n = 2.260776661 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 2.260776661 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 2.260776661 + 0.3333333334 n = 2.260776661 + 0.3333333334 Combine like terms: 2.260776661 + 0.3333333334 = 2.5941099944 n = 2.5941099944 Simplifying n = 2.5941099944

Subproblem 2

n + -0.3333333334 = -2.260776661 Simplifying n + -0.3333333334 = -2.260776661 Reorder the terms: -0.3333333334 + n = -2.260776661 Solving -0.3333333334 + n = -2.260776661 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -2.260776661 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -2.260776661 + 0.3333333334 n = -2.260776661 + 0.3333333334 Combine like terms: -2.260776661 + 0.3333333334 = -1.9274433276 n = -1.9274433276 Simplifying n = -1.9274433276

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.5941099944, -1.9274433276}

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