9n^2+27n-20=0

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Solution for 9n^2+27n-20=0 equation:


Simplifying
9n2 + 27n + -20 = 0

Reorder the terms:
-20 + 27n + 9n2 = 0

Solving
-20 + 27n + 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'.
-2.222222222 + 3n + n2 = 0

Move the constant term to the right:

Add '2.222222222' to each side of the equation.
-2.222222222 + 3n + 2.222222222 + n2 = 0 + 2.222222222

Reorder the terms:
-2.222222222 + 2.222222222 + 3n + n2 = 0 + 2.222222222

Combine like terms: -2.222222222 + 2.222222222 = 0.000000000
0.000000000 + 3n + n2 = 0 + 2.222222222
3n + n2 = 0 + 2.222222222

Combine like terms: 0 + 2.222222222 = 2.222222222
3n + n2 = 2.222222222

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 = 2.222222222 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 2.222222222 + 2.25

Combine like terms: 2.222222222 + 2.25 = 4.472222222
2.25 + 3n + n2 = 4.472222222

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 4.472222222

Calculate the square root of the right side: 2.114762923

Break this problem into two subproblems by setting 
(n + 1.5) equal to 2.114762923 and -2.114762923.

Subproblem 1

n + 1.5 = 2.114762923 Simplifying n + 1.5 = 2.114762923 Reorder the terms: 1.5 + n = 2.114762923 Solving 1.5 + n = 2.114762923 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 = 2.114762923 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.114762923 + -1.5 n = 2.114762923 + -1.5 Combine like terms: 2.114762923 + -1.5 = 0.614762923 n = 0.614762923 Simplifying n = 0.614762923

Subproblem 2

n + 1.5 = -2.114762923 Simplifying n + 1.5 = -2.114762923 Reorder the terms: 1.5 + n = -2.114762923 Solving 1.5 + n = -2.114762923 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 = -2.114762923 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.114762923 + -1.5 n = -2.114762923 + -1.5 Combine like terms: -2.114762923 + -1.5 = -3.614762923 n = -3.614762923 Simplifying n = -3.614762923

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.614762923, -3.614762923}

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