9n^2+18n-61=0

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Solution for 9n^2+18n-61=0 equation:


Simplifying
9n2 + 18n + -61 = 0

Reorder the terms:
-61 + 18n + 9n2 = 0

Solving
-61 + 18n + 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'.
-6.777777778 + 2n + n2 = 0

Move the constant term to the right:

Add '6.777777778' to each side of the equation.
-6.777777778 + 2n + 6.777777778 + n2 = 0 + 6.777777778

Reorder the terms:
-6.777777778 + 6.777777778 + 2n + n2 = 0 + 6.777777778

Combine like terms: -6.777777778 + 6.777777778 = 0.000000000
0.000000000 + 2n + n2 = 0 + 6.777777778
2n + n2 = 0 + 6.777777778

Combine like terms: 0 + 6.777777778 = 6.777777778
2n + n2 = 6.777777778

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 6.777777778 + 1

Reorder the terms:
1 + 2n + n2 = 6.777777778 + 1

Combine like terms: 6.777777778 + 1 = 7.777777778
1 + 2n + n2 = 7.777777778

Factor a perfect square on the left side:
(n + 1)(n + 1) = 7.777777778

Calculate the square root of the right side: 2.788866755

Break this problem into two subproblems by setting 
(n + 1) equal to 2.788866755 and -2.788866755.

Subproblem 1

n + 1 = 2.788866755 Simplifying n + 1 = 2.788866755 Reorder the terms: 1 + n = 2.788866755 Solving 1 + n = 2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 2.788866755 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.788866755 + -1 n = 2.788866755 + -1 Combine like terms: 2.788866755 + -1 = 1.788866755 n = 1.788866755 Simplifying n = 1.788866755

Subproblem 2

n + 1 = -2.788866755 Simplifying n + 1 = -2.788866755 Reorder the terms: 1 + n = -2.788866755 Solving 1 + n = -2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -2.788866755 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.788866755 + -1 n = -2.788866755 + -1 Combine like terms: -2.788866755 + -1 = -3.788866755 n = -3.788866755 Simplifying n = -3.788866755

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.788866755, -3.788866755}

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