3n^2-3n-8=12

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Solution for 3n^2-3n-8=12 equation:


Simplifying
3n2 + -3n + -8 = 12

Reorder the terms:
-8 + -3n + 3n2 = 12

Solving
-8 + -3n + 3n2 = 12

Solving for variable 'n'.

Reorder the terms:
-8 + -12 + -3n + 3n2 = 12 + -12

Combine like terms: -8 + -12 = -20
-20 + -3n + 3n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-20 + -3n + 3n2 = 0

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

Divide each side by '3'.
-6.666666667 + -1n + n2 = 0

Move the constant term to the right:

Add '6.666666667' to each side of the equation.
-6.666666667 + -1n + 6.666666667 + n2 = 0 + 6.666666667

Reorder the terms:
-6.666666667 + 6.666666667 + -1n + n2 = 0 + 6.666666667

Combine like terms: -6.666666667 + 6.666666667 = 0.000000000
0.000000000 + -1n + n2 = 0 + 6.666666667
-1n + n2 = 0 + 6.666666667

Combine like terms: 0 + 6.666666667 = 6.666666667
-1n + n2 = 6.666666667

The n term is -1n.  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.
-1n + 0.25 + n2 = 6.666666667 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 6.666666667 + 0.25

Combine like terms: 6.666666667 + 0.25 = 6.916666667
0.25 + -1n + n2 = 6.916666667

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(n + -0.5) equal to 2.62995564 and -2.62995564.

Subproblem 1

n + -0.5 = 2.62995564 Simplifying n + -0.5 = 2.62995564 Reorder the terms: -0.5 + n = 2.62995564 Solving -0.5 + n = 2.62995564 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 = 2.62995564 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 2.62995564 + 0.5 n = 2.62995564 + 0.5 Combine like terms: 2.62995564 + 0.5 = 3.12995564 n = 3.12995564 Simplifying n = 3.12995564

Subproblem 2

n + -0.5 = -2.62995564 Simplifying n + -0.5 = -2.62995564 Reorder the terms: -0.5 + n = -2.62995564 Solving -0.5 + n = -2.62995564 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 = -2.62995564 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -2.62995564 + 0.5 n = -2.62995564 + 0.5 Combine like terms: -2.62995564 + 0.5 = -2.12995564 n = -2.12995564 Simplifying n = -2.12995564

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.12995564, -2.12995564}

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