10*n^2+51*n-57500=0

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Solution for 10*n^2+51*n-57500=0 equation:


Simplifying
10n2 + 51n + -57500 = 0

Reorder the terms:
-57500 + 51n + 10n2 = 0

Solving
-57500 + 51n + 10n2 = 0

Solving for variable 'n'.

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

Divide each side by '10'.
-5750 + 5.1n + n2 = 0

Move the constant term to the right:

Add '5750' to each side of the equation.
-5750 + 5.1n + 5750 + n2 = 0 + 5750

Reorder the terms:
-5750 + 5750 + 5.1n + n2 = 0 + 5750

Combine like terms: -5750 + 5750 = 0
0 + 5.1n + n2 = 0 + 5750
5.1n + n2 = 0 + 5750

Combine like terms: 0 + 5750 = 5750
5.1n + n2 = 5750

The n term is 5.1n.  Take half its coefficient (2.55).
Square it (6.5025) and add it to both sides.

Add '6.5025' to each side of the equation.
5.1n + 6.5025 + n2 = 5750 + 6.5025

Reorder the terms:
6.5025 + 5.1n + n2 = 5750 + 6.5025

Combine like terms: 5750 + 6.5025 = 5756.5025
6.5025 + 5.1n + n2 = 5756.5025

Factor a perfect square on the left side:
(n + 2.55)(n + 2.55) = 5756.5025

Calculate the square root of the right side: 75.871618541

Break this problem into two subproblems by setting 
(n + 2.55) equal to 75.871618541 and -75.871618541.

Subproblem 1

n + 2.55 = 75.871618541 Simplifying n + 2.55 = 75.871618541 Reorder the terms: 2.55 + n = 75.871618541 Solving 2.55 + n = 75.871618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.55' to each side of the equation. 2.55 + -2.55 + n = 75.871618541 + -2.55 Combine like terms: 2.55 + -2.55 = 0.00 0.00 + n = 75.871618541 + -2.55 n = 75.871618541 + -2.55 Combine like terms: 75.871618541 + -2.55 = 73.321618541 n = 73.321618541 Simplifying n = 73.321618541

Subproblem 2

n + 2.55 = -75.871618541 Simplifying n + 2.55 = -75.871618541 Reorder the terms: 2.55 + n = -75.871618541 Solving 2.55 + n = -75.871618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.55' to each side of the equation. 2.55 + -2.55 + n = -75.871618541 + -2.55 Combine like terms: 2.55 + -2.55 = 0.00 0.00 + n = -75.871618541 + -2.55 n = -75.871618541 + -2.55 Combine like terms: -75.871618541 + -2.55 = -78.421618541 n = -78.421618541 Simplifying n = -78.421618541

Solution

The solution to the problem is based on the solutions from the subproblems. n = {73.321618541, -78.421618541}

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