3n^2+2n-9=-3

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


Simplifying
3n2 + 2n + -9 = -3

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

Solving
-9 + 2n + 3n2 = -3

Solving for variable 'n'.

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

Combine like terms: -9 + 3 = -6
-6 + 2n + 3n2 = -3 + 3

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

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

Divide each side by '3'.
-2 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.6666666667n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.6666666667n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.6666666667n + n2 = 0 + 2
0.6666666667n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.6666666667n + n2 = 2

The n term is 0.6666666667n.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667n + 0.1111111112 + n2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + 0.6666666667n + n2 = 2.1111111112

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1

n + 0.3333333334 = 1.452966315 Simplifying n + 0.3333333334 = 1.452966315 Reorder the terms: 0.3333333334 + n = 1.452966315 Solving 0.3333333334 + n = 1.452966315 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = 1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.452966315 + -0.3333333334 n = 1.452966315 + -0.3333333334 Combine like terms: 1.452966315 + -0.3333333334 = 1.1196329816 n = 1.1196329816 Simplifying n = 1.1196329816

Subproblem 2

n + 0.3333333334 = -1.452966315 Simplifying n + 0.3333333334 = -1.452966315 Reorder the terms: 0.3333333334 + n = -1.452966315 Solving 0.3333333334 + n = -1.452966315 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = -1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.452966315 + -0.3333333334 n = -1.452966315 + -0.3333333334 Combine like terms: -1.452966315 + -0.3333333334 = -1.7862996484 n = -1.7862996484 Simplifying n = -1.7862996484

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.1196329816, -1.7862996484}

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