8n^2-3n-16=24

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Solution for 8n^2-3n-16=24 equation:


Simplifying
8n2 + -3n + -16 = 24

Reorder the terms:
-16 + -3n + 8n2 = 24

Solving
-16 + -3n + 8n2 = 24

Solving for variable 'n'.

Reorder the terms:
-16 + -24 + -3n + 8n2 = 24 + -24

Combine like terms: -16 + -24 = -40
-40 + -3n + 8n2 = 24 + -24

Combine like terms: 24 + -24 = 0
-40 + -3n + 8n2 = 0

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

Divide each side by '8'.
-5 + -0.375n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.375n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.375n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.375n + n2 = 0 + 5
-0.375n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.375n + n2 = 5

The n term is -0.375n.  Take half its coefficient (-0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
-0.375n + 0.03515625 + n2 = 5 + 0.03515625

Reorder the terms:
0.03515625 + -0.375n + n2 = 5 + 0.03515625

Combine like terms: 5 + 0.03515625 = 5.03515625
0.03515625 + -0.375n + n2 = 5.03515625

Factor a perfect square on the left side:
(n + -0.1875)(n + -0.1875) = 5.03515625

Calculate the square root of the right side: 2.243915384

Break this problem into two subproblems by setting 
(n + -0.1875) equal to 2.243915384 and -2.243915384.

Subproblem 1

n + -0.1875 = 2.243915384 Simplifying n + -0.1875 = 2.243915384 Reorder the terms: -0.1875 + n = 2.243915384 Solving -0.1875 + n = 2.243915384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = 2.243915384 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 2.243915384 + 0.1875 n = 2.243915384 + 0.1875 Combine like terms: 2.243915384 + 0.1875 = 2.431415384 n = 2.431415384 Simplifying n = 2.431415384

Subproblem 2

n + -0.1875 = -2.243915384 Simplifying n + -0.1875 = -2.243915384 Reorder the terms: -0.1875 + n = -2.243915384 Solving -0.1875 + n = -2.243915384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = -2.243915384 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -2.243915384 + 0.1875 n = -2.243915384 + 0.1875 Combine like terms: -2.243915384 + 0.1875 = -2.056415384 n = -2.056415384 Simplifying n = -2.056415384

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.431415384, -2.056415384}

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