8n^2-12n=-1

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Solution for 8n^2-12n=-1 equation:


Simplifying
8n2 + -12n = -1

Reorder the terms:
-12n + 8n2 = -1

Solving
-12n + 8n2 = -1

Solving for variable 'n'.

Reorder the terms:
1 + -12n + 8n2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + -12n + 8n2 = 0

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

Divide each side by '8'.
0.125 + -1.5n + n2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + -1.5n + -0.125 + n2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + -1.5n + n2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + -1.5n + n2 = 0 + -0.125
-1.5n + n2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
-1.5n + n2 = -0.125

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = -0.125 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = -0.125 + 0.5625

Combine like terms: -0.125 + 0.5625 = 0.4375
0.5625 + -1.5n + n2 = 0.4375

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 0.4375

Calculate the square root of the right side: 0.661437828

Break this problem into two subproblems by setting 
(n + -0.75) equal to 0.661437828 and -0.661437828.

Subproblem 1

n + -0.75 = 0.661437828 Simplifying n + -0.75 = 0.661437828 Reorder the terms: -0.75 + n = 0.661437828 Solving -0.75 + n = 0.661437828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 0.661437828 + 0.75 n = 0.661437828 + 0.75 Combine like terms: 0.661437828 + 0.75 = 1.411437828 n = 1.411437828 Simplifying n = 1.411437828

Subproblem 2

n + -0.75 = -0.661437828 Simplifying n + -0.75 = -0.661437828 Reorder the terms: -0.75 + n = -0.661437828 Solving -0.75 + n = -0.661437828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -0.661437828 + 0.75 n = -0.661437828 + 0.75 Combine like terms: -0.661437828 + 0.75 = 0.088562172 n = 0.088562172 Simplifying n = 0.088562172

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.411437828, 0.088562172}

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