8n^2-n-6=0

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Solution for 8n^2-n-6=0 equation:


Simplifying
8n2 + -1n + -6 = 0

Reorder the terms:
-6 + -1n + 8n2 = 0

Solving
-6 + -1n + 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'.
-0.75 + -0.125n + n2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -0.125n + 0.75 + n2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -0.125n + n2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -0.125n + n2 = 0 + 0.75
-0.125n + n2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-0.125n + n2 = 0.75

The n term is -0.125n.  Take half its coefficient (-0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
-0.125n + 0.00390625 + n2 = 0.75 + 0.00390625

Reorder the terms:
0.00390625 + -0.125n + n2 = 0.75 + 0.00390625

Combine like terms: 0.75 + 0.00390625 = 0.75390625
0.00390625 + -0.125n + n2 = 0.75390625

Factor a perfect square on the left side:
(n + -0.0625)(n + -0.0625) = 0.75390625

Calculate the square root of the right side: 0.868277749

Break this problem into two subproblems by setting 
(n + -0.0625) equal to 0.868277749 and -0.868277749.

Subproblem 1

n + -0.0625 = 0.868277749 Simplifying n + -0.0625 = 0.868277749 Reorder the terms: -0.0625 + n = 0.868277749 Solving -0.0625 + n = 0.868277749 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = 0.868277749 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = 0.868277749 + 0.0625 n = 0.868277749 + 0.0625 Combine like terms: 0.868277749 + 0.0625 = 0.930777749 n = 0.930777749 Simplifying n = 0.930777749

Subproblem 2

n + -0.0625 = -0.868277749 Simplifying n + -0.0625 = -0.868277749 Reorder the terms: -0.0625 + n = -0.868277749 Solving -0.0625 + n = -0.868277749 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = -0.868277749 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = -0.868277749 + 0.0625 n = -0.868277749 + 0.0625 Combine like terms: -0.868277749 + 0.0625 = -0.805777749 n = -0.805777749 Simplifying n = -0.805777749

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.930777749, -0.805777749}

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