9n^2+24n+14=0

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Solution for 9n^2+24n+14=0 equation:


Simplifying
9n2 + 24n + 14 = 0

Reorder the terms:
14 + 24n + 9n2 = 0

Solving
14 + 24n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
1.555555556 + 2.666666667n + n2 = 0

Move the constant term to the right:

Add '-1.555555556' to each side of the equation.
1.555555556 + 2.666666667n + -1.555555556 + n2 = 0 + -1.555555556

Reorder the terms:
1.555555556 + -1.555555556 + 2.666666667n + n2 = 0 + -1.555555556

Combine like terms: 1.555555556 + -1.555555556 = 0.000000000
0.000000000 + 2.666666667n + n2 = 0 + -1.555555556
2.666666667n + n2 = 0 + -1.555555556

Combine like terms: 0 + -1.555555556 = -1.555555556
2.666666667n + n2 = -1.555555556

The n term is 2.666666667n.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667n + 1.777777780 + n2 = -1.555555556 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667n + n2 = -1.555555556 + 1.777777780

Combine like terms: -1.555555556 + 1.777777780 = 0.222222224
1.777777780 + 2.666666667n + n2 = 0.222222224

Factor a perfect square on the left side:
(n + 1.333333334)(n + 1.333333334) = 0.222222224

Calculate the square root of the right side: 0.471404523

Break this problem into two subproblems by setting 
(n + 1.333333334) equal to 0.471404523 and -0.471404523.

Subproblem 1

n + 1.333333334 = 0.471404523 Simplifying n + 1.333333334 = 0.471404523 Reorder the terms: 1.333333334 + n = 0.471404523 Solving 1.333333334 + n = 0.471404523 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = 0.471404523 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 0.471404523 + -1.333333334 n = 0.471404523 + -1.333333334 Combine like terms: 0.471404523 + -1.333333334 = -0.861928811 n = -0.861928811 Simplifying n = -0.861928811

Subproblem 2

n + 1.333333334 = -0.471404523 Simplifying n + 1.333333334 = -0.471404523 Reorder the terms: 1.333333334 + n = -0.471404523 Solving 1.333333334 + n = -0.471404523 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = -0.471404523 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -0.471404523 + -1.333333334 n = -0.471404523 + -1.333333334 Combine like terms: -0.471404523 + -1.333333334 = -1.804737857 n = -1.804737857 Simplifying n = -1.804737857

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.861928811, -1.804737857}

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