8n^2-9n-18=0

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Solution for 8n^2-9n-18=0 equation:


Simplifying
8n2 + -9n + -18 = 0

Reorder the terms:
-18 + -9n + 8n2 = 0

Solving
-18 + -9n + 8n2 = 0

Solving for variable 'n'.

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

Divide each side by '8'.
-2.25 + -1.125n + n2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -1.125n + 2.25 + n2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -1.125n + n2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -1.125n + n2 = 0 + 2.25
-1.125n + n2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-1.125n + n2 = 2.25

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 = 2.25 + 0.31640625

Reorder the terms:
0.31640625 + -1.125n + n2 = 2.25 + 0.31640625

Combine like terms: 2.25 + 0.31640625 = 2.56640625
0.31640625 + -1.125n + n2 = 2.56640625

Factor a perfect square on the left side:
(n + -0.5625)(n + -0.5625) = 2.56640625

Calculate the square root of the right side: 1.602000702

Break this problem into two subproblems by setting 
(n + -0.5625) equal to 1.602000702 and -1.602000702.

Subproblem 1

n + -0.5625 = 1.602000702 Simplifying n + -0.5625 = 1.602000702 Reorder the terms: -0.5625 + n = 1.602000702 Solving -0.5625 + n = 1.602000702 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 = 1.602000702 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = 1.602000702 + 0.5625 n = 1.602000702 + 0.5625 Combine like terms: 1.602000702 + 0.5625 = 2.164500702 n = 2.164500702 Simplifying n = 2.164500702

Subproblem 2

n + -0.5625 = -1.602000702 Simplifying n + -0.5625 = -1.602000702 Reorder the terms: -0.5625 + n = -1.602000702 Solving -0.5625 + n = -1.602000702 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 = -1.602000702 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = -1.602000702 + 0.5625 n = -1.602000702 + 0.5625 Combine like terms: -1.602000702 + 0.5625 = -1.039500702 n = -1.039500702 Simplifying n = -1.039500702

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.164500702, -1.039500702}

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