3n^2-6n=8

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Solution for 3n^2-6n=8 equation:


Simplifying
3n2 + -6n = 8

Reorder the terms:
-6n + 3n2 = 8

Solving
-6n + 3n2 = 8

Solving for variable 'n'.

Reorder the terms:
-8 + -6n + 3n2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + -6n + 3n2 = 0

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

Divide each side by '3'.
-2.666666667 + -2n + n2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -2n + 2.666666667 + n2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -2n + n2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -2n + n2 = 0 + 2.666666667
-2n + n2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-2n + n2 = 2.666666667

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 2.666666667 + 1

Reorder the terms:
1 + -2n + n2 = 2.666666667 + 1

Combine like terms: 2.666666667 + 1 = 3.666666667
1 + -2n + n2 = 3.666666667

Factor a perfect square on the left side:
(n + -1)(n + -1) = 3.666666667

Calculate the square root of the right side: 1.914854216

Break this problem into two subproblems by setting 
(n + -1) equal to 1.914854216 and -1.914854216.

Subproblem 1

n + -1 = 1.914854216 Simplifying n + -1 = 1.914854216 Reorder the terms: -1 + n = 1.914854216 Solving -1 + n = 1.914854216 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.914854216 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.914854216 + 1 n = 1.914854216 + 1 Combine like terms: 1.914854216 + 1 = 2.914854216 n = 2.914854216 Simplifying n = 2.914854216

Subproblem 2

n + -1 = -1.914854216 Simplifying n + -1 = -1.914854216 Reorder the terms: -1 + n = -1.914854216 Solving -1 + n = -1.914854216 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.914854216 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.914854216 + 1 n = -1.914854216 + 1 Combine like terms: -1.914854216 + 1 = -0.914854216 n = -0.914854216 Simplifying n = -0.914854216

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.914854216, -0.914854216}

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