8n^2-19n-24=0

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Solution for 8n^2-19n-24=0 equation:


Simplifying
8n2 + -19n + -24 = 0

Reorder the terms:
-24 + -19n + 8n2 = 0

Solving
-24 + -19n + 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 + -2.375n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -2.375n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + -2.375n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -2.375n + n2 = 0 + 3
-2.375n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
-2.375n + n2 = 3

The n term is -2.375n.  Take half its coefficient (-1.1875).
Square it (1.41015625) and add it to both sides.

Add '1.41015625' to each side of the equation.
-2.375n + 1.41015625 + n2 = 3 + 1.41015625

Reorder the terms:
1.41015625 + -2.375n + n2 = 3 + 1.41015625

Combine like terms: 3 + 1.41015625 = 4.41015625
1.41015625 + -2.375n + n2 = 4.41015625

Factor a perfect square on the left side:
(n + -1.1875)(n + -1.1875) = 4.41015625

Calculate the square root of the right side: 2.100037202

Break this problem into two subproblems by setting 
(n + -1.1875) equal to 2.100037202 and -2.100037202.

Subproblem 1

n + -1.1875 = 2.100037202 Simplifying n + -1.1875 = 2.100037202 Reorder the terms: -1.1875 + n = 2.100037202 Solving -1.1875 + n = 2.100037202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = 2.100037202 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = 2.100037202 + 1.1875 n = 2.100037202 + 1.1875 Combine like terms: 2.100037202 + 1.1875 = 3.287537202 n = 3.287537202 Simplifying n = 3.287537202

Subproblem 2

n + -1.1875 = -2.100037202 Simplifying n + -1.1875 = -2.100037202 Reorder the terms: -1.1875 + n = -2.100037202 Solving -1.1875 + n = -2.100037202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = -2.100037202 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = -2.100037202 + 1.1875 n = -2.100037202 + 1.1875 Combine like terms: -2.100037202 + 1.1875 = -0.912537202 n = -0.912537202 Simplifying n = -0.912537202

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.287537202, -0.912537202}

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