8n^2-4n-18=9

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Solution for 8n^2-4n-18=9 equation:


Simplifying
8n2 + -4n + -18 = 9

Reorder the terms:
-18 + -4n + 8n2 = 9

Solving
-18 + -4n + 8n2 = 9

Solving for variable 'n'.

Reorder the terms:
-18 + -9 + -4n + 8n2 = 9 + -9

Combine like terms: -18 + -9 = -27
-27 + -4n + 8n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-27 + -4n + 8n2 = 0

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

Divide each side by '8'.
-3.375 + -0.5n + n2 = 0

Move the constant term to the right:

Add '3.375' to each side of the equation.
-3.375 + -0.5n + 3.375 + n2 = 0 + 3.375

Reorder the terms:
-3.375 + 3.375 + -0.5n + n2 = 0 + 3.375

Combine like terms: -3.375 + 3.375 = 0.000
0.000 + -0.5n + n2 = 0 + 3.375
-0.5n + n2 = 0 + 3.375

Combine like terms: 0 + 3.375 = 3.375
-0.5n + n2 = 3.375

The n term is -0.5n.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5n + 0.0625 + n2 = 3.375 + 0.0625

Reorder the terms:
0.0625 + -0.5n + n2 = 3.375 + 0.0625

Combine like terms: 3.375 + 0.0625 = 3.4375
0.0625 + -0.5n + n2 = 3.4375

Factor a perfect square on the left side:
(n + -0.25)(n + -0.25) = 3.4375

Calculate the square root of the right side: 1.854049622

Break this problem into two subproblems by setting 
(n + -0.25) equal to 1.854049622 and -1.854049622.

Subproblem 1

n + -0.25 = 1.854049622 Simplifying n + -0.25 = 1.854049622 Reorder the terms: -0.25 + n = 1.854049622 Solving -0.25 + n = 1.854049622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 1.854049622 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 1.854049622 + 0.25 n = 1.854049622 + 0.25 Combine like terms: 1.854049622 + 0.25 = 2.104049622 n = 2.104049622 Simplifying n = 2.104049622

Subproblem 2

n + -0.25 = -1.854049622 Simplifying n + -0.25 = -1.854049622 Reorder the terms: -0.25 + n = -1.854049622 Solving -0.25 + n = -1.854049622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -1.854049622 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -1.854049622 + 0.25 n = -1.854049622 + 0.25 Combine like terms: -1.854049622 + 0.25 = -1.604049622 n = -1.604049622 Simplifying n = -1.604049622

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.104049622, -1.604049622}

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