9n^2-6n-26=0

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Solution for 9n^2-6n-26=0 equation:


Simplifying
9n2 + -6n + -26 = 0

Reorder the terms:
-26 + -6n + 9n2 = 0

Solving
-26 + -6n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
-2.888888889 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '2.888888889' to each side of the equation.
-2.888888889 + -0.6666666667n + 2.888888889 + n2 = 0 + 2.888888889

Reorder the terms:
-2.888888889 + 2.888888889 + -0.6666666667n + n2 = 0 + 2.888888889

Combine like terms: -2.888888889 + 2.888888889 = 0.000000000
0.000000000 + -0.6666666667n + n2 = 0 + 2.888888889
-0.6666666667n + n2 = 0 + 2.888888889

Combine like terms: 0 + 2.888888889 = 2.888888889
-0.6666666667n + n2 = 2.888888889

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 = 2.888888889 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 2.888888889 + 0.1111111112

Combine like terms: 2.888888889 + 0.1111111112 = 3.0000000002
0.1111111112 + -0.6666666667n + n2 = 3.0000000002

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 3.0000000002

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + -0.3333333334 = 1.732050808 Simplifying n + -0.3333333334 = 1.732050808 Reorder the terms: -0.3333333334 + n = 1.732050808 Solving -0.3333333334 + n = 1.732050808 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 = 1.732050808 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 1.732050808 + 0.3333333334 n = 1.732050808 + 0.3333333334 Combine like terms: 1.732050808 + 0.3333333334 = 2.0653841414 n = 2.0653841414 Simplifying n = 2.0653841414

Subproblem 2

n + -0.3333333334 = -1.732050808 Simplifying n + -0.3333333334 = -1.732050808 Reorder the terms: -0.3333333334 + n = -1.732050808 Solving -0.3333333334 + n = -1.732050808 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 = -1.732050808 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -1.732050808 + 0.3333333334 n = -1.732050808 + 0.3333333334 Combine like terms: -1.732050808 + 0.3333333334 = -1.3987174746 n = -1.3987174746 Simplifying n = -1.3987174746

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.0653841414, -1.3987174746}

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