8n^2-6n-21=0

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Solution for 8n^2-6n-21=0 equation:


Simplifying
8n2 + -6n + -21 = 0

Reorder the terms:
-21 + -6n + 8n2 = 0

Solving
-21 + -6n + 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'.
-2.625 + -0.75n + n2 = 0

Move the constant term to the right:

Add '2.625' to each side of the equation.
-2.625 + -0.75n + 2.625 + n2 = 0 + 2.625

Reorder the terms:
-2.625 + 2.625 + -0.75n + n2 = 0 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + -0.75n + n2 = 0 + 2.625
-0.75n + n2 = 0 + 2.625

Combine like terms: 0 + 2.625 = 2.625
-0.75n + n2 = 2.625

The n term is -0.75n.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75n + 0.140625 + n2 = 2.625 + 0.140625

Reorder the terms:
0.140625 + -0.75n + n2 = 2.625 + 0.140625

Combine like terms: 2.625 + 0.140625 = 2.765625
0.140625 + -0.75n + n2 = 2.765625

Factor a perfect square on the left side:
(n + -0.375)(n + -0.375) = 2.765625

Calculate the square root of the right side: 1.663016837

Break this problem into two subproblems by setting 
(n + -0.375) equal to 1.663016837 and -1.663016837.

Subproblem 1

n + -0.375 = 1.663016837 Simplifying n + -0.375 = 1.663016837 Reorder the terms: -0.375 + n = 1.663016837 Solving -0.375 + n = 1.663016837 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 1.663016837 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 1.663016837 + 0.375 n = 1.663016837 + 0.375 Combine like terms: 1.663016837 + 0.375 = 2.038016837 n = 2.038016837 Simplifying n = 2.038016837

Subproblem 2

n + -0.375 = -1.663016837 Simplifying n + -0.375 = -1.663016837 Reorder the terms: -0.375 + n = -1.663016837 Solving -0.375 + n = -1.663016837 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -1.663016837 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -1.663016837 + 0.375 n = -1.663016837 + 0.375 Combine like terms: -1.663016837 + 0.375 = -1.288016837 n = -1.288016837 Simplifying n = -1.288016837

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.038016837, -1.288016837}

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