3n^2-5n-6=9

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Solution for 3n^2-5n-6=9 equation:


Simplifying
3n2 + -5n + -6 = 9

Reorder the terms:
-6 + -5n + 3n2 = 9

Solving
-6 + -5n + 3n2 = 9

Solving for variable 'n'.

Reorder the terms:
-6 + -9 + -5n + 3n2 = 9 + -9

Combine like terms: -6 + -9 = -15
-15 + -5n + 3n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-15 + -5n + 3n2 = 0

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

Divide each side by '3'.
-5 + -1.666666667n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1.666666667n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -1.666666667n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1.666666667n + n2 = 0 + 5
-1.666666667n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1.666666667n + n2 = 5

The n term is -1.666666667n.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667n + 0.6944444447 + n2 = 5 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667n + n2 = 5 + 0.6944444447

Combine like terms: 5 + 0.6944444447 = 5.6944444447
0.6944444447 + -1.666666667n + n2 = 5.6944444447

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 5.6944444447

Calculate the square root of the right side: 2.386303511

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 2.386303511 and -2.386303511.

Subproblem 1

n + -0.8333333335 = 2.386303511 Simplifying n + -0.8333333335 = 2.386303511 Reorder the terms: -0.8333333335 + n = 2.386303511 Solving -0.8333333335 + n = 2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 2.386303511 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 2.386303511 + 0.8333333335 n = 2.386303511 + 0.8333333335 Combine like terms: 2.386303511 + 0.8333333335 = 3.2196368445 n = 3.2196368445 Simplifying n = 3.2196368445

Subproblem 2

n + -0.8333333335 = -2.386303511 Simplifying n + -0.8333333335 = -2.386303511 Reorder the terms: -0.8333333335 + n = -2.386303511 Solving -0.8333333335 + n = -2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -2.386303511 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -2.386303511 + 0.8333333335 n = -2.386303511 + 0.8333333335 Combine like terms: -2.386303511 + 0.8333333335 = -1.5529701775 n = -1.5529701775 Simplifying n = -1.5529701775

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.2196368445, -1.5529701775}

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