3n^2-9n=2

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


Simplifying
3n2 + -9n = 2

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

Solving
-9n + 3n2 = 2

Solving for variable 'n'.

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

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

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

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

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3n + 0.6666666667 + n2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3n + n2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3n + n2 = 0 + 0.6666666667
-3n + n2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3n + n2 = 0.6666666667

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 0.6666666667 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 0.6666666667 + 2.25

Combine like terms: 0.6666666667 + 2.25 = 2.9166666667
2.25 + -3n + n2 = 2.9166666667

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 2.9166666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(n + -1.5) equal to 1.707825128 and -1.707825128.

Subproblem 1

n + -1.5 = 1.707825128 Simplifying n + -1.5 = 1.707825128 Reorder the terms: -1.5 + n = 1.707825128 Solving -1.5 + n = 1.707825128 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.707825128 + 1.5 n = 1.707825128 + 1.5 Combine like terms: 1.707825128 + 1.5 = 3.207825128 n = 3.207825128 Simplifying n = 3.207825128

Subproblem 2

n + -1.5 = -1.707825128 Simplifying n + -1.5 = -1.707825128 Reorder the terms: -1.5 + n = -1.707825128 Solving -1.5 + n = -1.707825128 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.707825128 + 1.5 n = -1.707825128 + 1.5 Combine like terms: -1.707825128 + 1.5 = -0.207825128 n = -0.207825128 Simplifying n = -0.207825128

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.207825128, -0.207825128}

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