8n^2-51n-56=0

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Solution for 8n^2-51n-56=0 equation:


Simplifying
8n2 + -51n + -56 = 0

Reorder the terms:
-56 + -51n + 8n2 = 0

Solving
-56 + -51n + 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'.
-7 + -6.375n + n2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -6.375n + 7 + n2 = 0 + 7

Reorder the terms:
-7 + 7 + -6.375n + n2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -6.375n + n2 = 0 + 7
-6.375n + n2 = 0 + 7

Combine like terms: 0 + 7 = 7
-6.375n + n2 = 7

The n term is -6.375n.  Take half its coefficient (-3.1875).
Square it (10.16015625) and add it to both sides.

Add '10.16015625' to each side of the equation.
-6.375n + 10.16015625 + n2 = 7 + 10.16015625

Reorder the terms:
10.16015625 + -6.375n + n2 = 7 + 10.16015625

Combine like terms: 7 + 10.16015625 = 17.16015625
10.16015625 + -6.375n + n2 = 17.16015625

Factor a perfect square on the left side:
(n + -3.1875)(n + -3.1875) = 17.16015625

Calculate the square root of the right side: 4.142481895

Break this problem into two subproblems by setting 
(n + -3.1875) equal to 4.142481895 and -4.142481895.

Subproblem 1

n + -3.1875 = 4.142481895 Simplifying n + -3.1875 = 4.142481895 Reorder the terms: -3.1875 + n = 4.142481895 Solving -3.1875 + n = 4.142481895 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.1875' to each side of the equation. -3.1875 + 3.1875 + n = 4.142481895 + 3.1875 Combine like terms: -3.1875 + 3.1875 = 0.0000 0.0000 + n = 4.142481895 + 3.1875 n = 4.142481895 + 3.1875 Combine like terms: 4.142481895 + 3.1875 = 7.329981895 n = 7.329981895 Simplifying n = 7.329981895

Subproblem 2

n + -3.1875 = -4.142481895 Simplifying n + -3.1875 = -4.142481895 Reorder the terms: -3.1875 + n = -4.142481895 Solving -3.1875 + n = -4.142481895 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.1875' to each side of the equation. -3.1875 + 3.1875 + n = -4.142481895 + 3.1875 Combine like terms: -3.1875 + 3.1875 = 0.0000 0.0000 + n = -4.142481895 + 3.1875 n = -4.142481895 + 3.1875 Combine like terms: -4.142481895 + 3.1875 = -0.954981895 n = -0.954981895 Simplifying n = -0.954981895

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.329981895, -0.954981895}

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