8n^2+9n-18=2

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Solution for 8n^2+9n-18=2 equation:


Simplifying
8n2 + 9n + -18 = 2

Reorder the terms:
-18 + 9n + 8n2 = 2

Solving
-18 + 9n + 8n2 = 2

Solving for variable 'n'.

Reorder the terms:
-18 + -2 + 9n + 8n2 = 2 + -2

Combine like terms: -18 + -2 = -20
-20 + 9n + 8n2 = 2 + -2

Combine like terms: 2 + -2 = 0
-20 + 9n + 8n2 = 0

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

Divide each side by '8'.
-2.5 + 1.125n + n2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 1.125n + 2.5 + n2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 1.125n + n2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 1.125n + n2 = 0 + 2.5
1.125n + n2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
1.125n + n2 = 2.5

The n term is 1.125n.  Take half its coefficient (0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
1.125n + 0.31640625 + n2 = 2.5 + 0.31640625

Reorder the terms:
0.31640625 + 1.125n + n2 = 2.5 + 0.31640625

Combine like terms: 2.5 + 0.31640625 = 2.81640625
0.31640625 + 1.125n + n2 = 2.81640625

Factor a perfect square on the left side:
(n + 0.5625)(n + 0.5625) = 2.81640625

Calculate the square root of the right side: 1.678215198

Break this problem into two subproblems by setting 
(n + 0.5625) equal to 1.678215198 and -1.678215198.

Subproblem 1

n + 0.5625 = 1.678215198 Simplifying n + 0.5625 = 1.678215198 Reorder the terms: 0.5625 + n = 1.678215198 Solving 0.5625 + n = 1.678215198 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = 1.678215198 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = 1.678215198 + -0.5625 n = 1.678215198 + -0.5625 Combine like terms: 1.678215198 + -0.5625 = 1.115715198 n = 1.115715198 Simplifying n = 1.115715198

Subproblem 2

n + 0.5625 = -1.678215198 Simplifying n + 0.5625 = -1.678215198 Reorder the terms: 0.5625 + n = -1.678215198 Solving 0.5625 + n = -1.678215198 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = -1.678215198 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = -1.678215198 + -0.5625 n = -1.678215198 + -0.5625 Combine like terms: -1.678215198 + -0.5625 = -2.240715198 n = -2.240715198 Simplifying n = -2.240715198

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.115715198, -2.240715198}

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