8n^2+4n-21=0

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Solution for 8n^2+4n-21=0 equation:


Simplifying
8n2 + 4n + -21 = 0

Reorder the terms:
-21 + 4n + 8n2 = 0

Solving
-21 + 4n + 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 + 0.5n + n2 = 0

Move the constant term to the right:

Add '2.625' to each side of the equation.
-2.625 + 0.5n + 2.625 + n2 = 0 + 2.625

Reorder the terms:
-2.625 + 2.625 + 0.5n + n2 = 0 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + 0.5n + n2 = 0 + 2.625
0.5n + n2 = 0 + 2.625

Combine like terms: 0 + 2.625 = 2.625
0.5n + n2 = 2.625

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 = 2.625 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 2.625 + 0.0625

Combine like terms: 2.625 + 0.0625 = 2.6875
0.0625 + 0.5n + n2 = 2.6875

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 2.6875

Calculate the square root of the right side: 1.639359631

Break this problem into two subproblems by setting 
(n + 0.25) equal to 1.639359631 and -1.639359631.

Subproblem 1

n + 0.25 = 1.639359631 Simplifying n + 0.25 = 1.639359631 Reorder the terms: 0.25 + n = 1.639359631 Solving 0.25 + n = 1.639359631 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 = 1.639359631 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.639359631 + -0.25 n = 1.639359631 + -0.25 Combine like terms: 1.639359631 + -0.25 = 1.389359631 n = 1.389359631 Simplifying n = 1.389359631

Subproblem 2

n + 0.25 = -1.639359631 Simplifying n + 0.25 = -1.639359631 Reorder the terms: 0.25 + n = -1.639359631 Solving 0.25 + n = -1.639359631 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 = -1.639359631 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.639359631 + -0.25 n = -1.639359631 + -0.25 Combine like terms: -1.639359631 + -0.25 = -1.889359631 n = -1.889359631 Simplifying n = -1.889359631

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.389359631, -1.889359631}

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