4n^2-21n-8=0

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


Simplifying
4n2 + -21n + -8 = 0

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

Solving
-8 + -21n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-2 + -5.25n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -5.25n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -5.25n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -5.25n + n2 = 0 + 2
-5.25n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-5.25n + n2 = 2

The n term is -5.25n.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25n + 6.890625 + n2 = 2 + 6.890625

Reorder the terms:
6.890625 + -5.25n + n2 = 2 + 6.890625

Combine like terms: 2 + 6.890625 = 8.890625
6.890625 + -5.25n + n2 = 8.890625

Factor a perfect square on the left side:
(n + -2.625)(n + -2.625) = 8.890625

Calculate the square root of the right side: 2.98171511

Break this problem into two subproblems by setting 
(n + -2.625) equal to 2.98171511 and -2.98171511.

Subproblem 1

n + -2.625 = 2.98171511 Simplifying n + -2.625 = 2.98171511 Reorder the terms: -2.625 + n = 2.98171511 Solving -2.625 + n = 2.98171511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = 2.98171511 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = 2.98171511 + 2.625 n = 2.98171511 + 2.625 Combine like terms: 2.98171511 + 2.625 = 5.60671511 n = 5.60671511 Simplifying n = 5.60671511

Subproblem 2

n + -2.625 = -2.98171511 Simplifying n + -2.625 = -2.98171511 Reorder the terms: -2.625 + n = -2.98171511 Solving -2.625 + n = -2.98171511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = -2.98171511 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = -2.98171511 + 2.625 n = -2.98171511 + 2.625 Combine like terms: -2.98171511 + 2.625 = -0.35671511 n = -0.35671511 Simplifying n = -0.35671511

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.60671511, -0.35671511}

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