2n^2-6n=-3

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


Simplifying
2n2 + -6n = -3

Reorder the terms:
-6n + 2n2 = -3

Solving
-6n + 2n2 = -3

Solving for variable 'n'.

Reorder the terms:
3 + -6n + 2n2 = -3 + 3

Combine like terms: -3 + 3 = 0
3 + -6n + 2n2 = 0

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

Divide each side by '2'.
1.5 + -3n + n2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3n + -1.5 + n2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3n + n2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3n + n2 = 0 + -1.5
-3n + n2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3n + n2 = -1.5

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 = -1.5 + 2.25

Reorder the terms:
2.25 + -3n + n2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3n + n2 = 0.75

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(n + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

n + -1.5 = 0.866025404 Simplifying n + -1.5 = 0.866025404 Reorder the terms: -1.5 + n = 0.866025404 Solving -1.5 + n = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 0.866025404 + 1.5 n = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 n = 2.366025404 Simplifying n = 2.366025404

Subproblem 2

n + -1.5 = -0.866025404 Simplifying n + -1.5 = -0.866025404 Reorder the terms: -1.5 + n = -0.866025404 Solving -1.5 + n = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -0.866025404 + 1.5 n = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 n = 0.633974596 Simplifying n = 0.633974596

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.366025404, 0.633974596}

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