8n^2-5n-10=0

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Solution for 8n^2-5n-10=0 equation:


Simplifying
8n2 + -5n + -10 = 0

Reorder the terms:
-10 + -5n + 8n2 = 0

Solving
-10 + -5n + 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'.
-1.25 + -0.625n + n2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -0.625n + 1.25 + n2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -0.625n + n2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -0.625n + n2 = 0 + 1.25
-0.625n + n2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-0.625n + n2 = 1.25

The n term is -0.625n.  Take half its coefficient (-0.3125).
Square it (0.09765625) and add it to both sides.

Add '0.09765625' to each side of the equation.
-0.625n + 0.09765625 + n2 = 1.25 + 0.09765625

Reorder the terms:
0.09765625 + -0.625n + n2 = 1.25 + 0.09765625

Combine like terms: 1.25 + 0.09765625 = 1.34765625
0.09765625 + -0.625n + n2 = 1.34765625

Factor a perfect square on the left side:
(n + -0.3125)(n + -0.3125) = 1.34765625

Calculate the square root of the right side: 1.160885976

Break this problem into two subproblems by setting 
(n + -0.3125) equal to 1.160885976 and -1.160885976.

Subproblem 1

n + -0.3125 = 1.160885976 Simplifying n + -0.3125 = 1.160885976 Reorder the terms: -0.3125 + n = 1.160885976 Solving -0.3125 + n = 1.160885976 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + n = 1.160885976 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + n = 1.160885976 + 0.3125 n = 1.160885976 + 0.3125 Combine like terms: 1.160885976 + 0.3125 = 1.473385976 n = 1.473385976 Simplifying n = 1.473385976

Subproblem 2

n + -0.3125 = -1.160885976 Simplifying n + -0.3125 = -1.160885976 Reorder the terms: -0.3125 + n = -1.160885976 Solving -0.3125 + n = -1.160885976 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + n = -1.160885976 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + n = -1.160885976 + 0.3125 n = -1.160885976 + 0.3125 Combine like terms: -1.160885976 + 0.3125 = -0.848385976 n = -0.848385976 Simplifying n = -0.848385976

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.473385976, -0.848385976}

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