2n^2+6n-9=0

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Solution for 2n^2+6n-9=0 equation:


Simplifying
2n2 + 6n + -9 = 0

Reorder the terms:
-9 + 6n + 2n2 = 0

Solving
-9 + 6n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
-4.5 + 3n + n2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 3n + 4.5 + n2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 3n + n2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 3n + n2 = 0 + 4.5
3n + n2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
3n + n2 = 4.5

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 = 4.5 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 4.5 + 2.25

Combine like terms: 4.5 + 2.25 = 6.75
2.25 + 3n + n2 = 6.75

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(n + 1.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

n + 1.5 = 2.598076211 Simplifying n + 1.5 = 2.598076211 Reorder the terms: 1.5 + n = 2.598076211 Solving 1.5 + n = 2.598076211 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 = 2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.598076211 + -1.5 n = 2.598076211 + -1.5 Combine like terms: 2.598076211 + -1.5 = 1.098076211 n = 1.098076211 Simplifying n = 1.098076211

Subproblem 2

n + 1.5 = -2.598076211 Simplifying n + 1.5 = -2.598076211 Reorder the terms: 1.5 + n = -2.598076211 Solving 1.5 + n = -2.598076211 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 = -2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.598076211 + -1.5 n = -2.598076211 + -1.5 Combine like terms: -2.598076211 + -1.5 = -4.098076211 n = -4.098076211 Simplifying n = -4.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.098076211, -4.098076211}

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