8n^2-21n-24=0

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Solution for 8n^2-21n-24=0 equation:


Simplifying
8n2 + -21n + -24 = 0

Reorder the terms:
-24 + -21n + 8n2 = 0

Solving
-24 + -21n + 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'.
-3 + -2.625n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -2.625n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + -2.625n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -2.625n + n2 = 0 + 3
-2.625n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
-2.625n + n2 = 3

The n term is -2.625n.  Take half its coefficient (-1.3125).
Square it (1.72265625) and add it to both sides.

Add '1.72265625' to each side of the equation.
-2.625n + 1.72265625 + n2 = 3 + 1.72265625

Reorder the terms:
1.72265625 + -2.625n + n2 = 3 + 1.72265625

Combine like terms: 3 + 1.72265625 = 4.72265625
1.72265625 + -2.625n + n2 = 4.72265625

Factor a perfect square on the left side:
(n + -1.3125)(n + -1.3125) = 4.72265625

Calculate the square root of the right side: 2.173167331

Break this problem into two subproblems by setting 
(n + -1.3125) equal to 2.173167331 and -2.173167331.

Subproblem 1

n + -1.3125 = 2.173167331 Simplifying n + -1.3125 = 2.173167331 Reorder the terms: -1.3125 + n = 2.173167331 Solving -1.3125 + n = 2.173167331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + n = 2.173167331 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + n = 2.173167331 + 1.3125 n = 2.173167331 + 1.3125 Combine like terms: 2.173167331 + 1.3125 = 3.485667331 n = 3.485667331 Simplifying n = 3.485667331

Subproblem 2

n + -1.3125 = -2.173167331 Simplifying n + -1.3125 = -2.173167331 Reorder the terms: -1.3125 + n = -2.173167331 Solving -1.3125 + n = -2.173167331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + n = -2.173167331 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + n = -2.173167331 + 1.3125 n = -2.173167331 + 1.3125 Combine like terms: -2.173167331 + 1.3125 = -0.860667331 n = -0.860667331 Simplifying n = -0.860667331

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.485667331, -0.860667331}

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