8n^2+7n-16=0

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Solution for 8n^2+7n-16=0 equation:


Simplifying
8n2 + 7n + -16 = 0

Reorder the terms:
-16 + 7n + 8n2 = 0

Solving
-16 + 7n + 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 + 0.875n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.875n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.875n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.875n + n2 = 0 + 2
0.875n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.875n + n2 = 2

The n term is 0.875n.  Take half its coefficient (0.4375).
Square it (0.19140625) and add it to both sides.

Add '0.19140625' to each side of the equation.
0.875n + 0.19140625 + n2 = 2 + 0.19140625

Reorder the terms:
0.19140625 + 0.875n + n2 = 2 + 0.19140625

Combine like terms: 2 + 0.19140625 = 2.19140625
0.19140625 + 0.875n + n2 = 2.19140625

Factor a perfect square on the left side:
(n + 0.4375)(n + 0.4375) = 2.19140625

Calculate the square root of the right side: 1.48033991

Break this problem into two subproblems by setting 
(n + 0.4375) equal to 1.48033991 and -1.48033991.

Subproblem 1

n + 0.4375 = 1.48033991 Simplifying n + 0.4375 = 1.48033991 Reorder the terms: 0.4375 + n = 1.48033991 Solving 0.4375 + n = 1.48033991 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + n = 1.48033991 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = 1.48033991 + -0.4375 n = 1.48033991 + -0.4375 Combine like terms: 1.48033991 + -0.4375 = 1.04283991 n = 1.04283991 Simplifying n = 1.04283991

Subproblem 2

n + 0.4375 = -1.48033991 Simplifying n + 0.4375 = -1.48033991 Reorder the terms: 0.4375 + n = -1.48033991 Solving 0.4375 + n = -1.48033991 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + n = -1.48033991 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = -1.48033991 + -0.4375 n = -1.48033991 + -0.4375 Combine like terms: -1.48033991 + -0.4375 = -1.91783991 n = -1.91783991 Simplifying n = -1.91783991

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.04283991, -1.91783991}

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