10n^2+80n-262.5=0

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Solution for 10n^2+80n-262.5=0 equation:


Simplifying
10n2 + 80n + -262.5 = 0

Reorder the terms:
-262.5 + 80n + 10n2 = 0

Solving
-262.5 + 80n + 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'.
-26.25 + 8n + n2 = 0

Move the constant term to the right:

Add '26.25' to each side of the equation.
-26.25 + 8n + 26.25 + n2 = 0 + 26.25

Reorder the terms:
-26.25 + 26.25 + 8n + n2 = 0 + 26.25

Combine like terms: -26.25 + 26.25 = 0.00
0.00 + 8n + n2 = 0 + 26.25
8n + n2 = 0 + 26.25

Combine like terms: 0 + 26.25 = 26.25
8n + n2 = 26.25

The n term is 8n.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8n + 16 + n2 = 26.25 + 16

Reorder the terms:
16 + 8n + n2 = 26.25 + 16

Combine like terms: 26.25 + 16 = 42.25
16 + 8n + n2 = 42.25

Factor a perfect square on the left side:
(n + 4)(n + 4) = 42.25

Calculate the square root of the right side: 6.5

Break this problem into two subproblems by setting 
(n + 4) equal to 6.5 and -6.5.

Subproblem 1

n + 4 = 6.5 Simplifying n + 4 = 6.5 Reorder the terms: 4 + n = 6.5 Solving 4 + n = 6.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = 6.5 + -4 Combine like terms: 4 + -4 = 0 0 + n = 6.5 + -4 n = 6.5 + -4 Combine like terms: 6.5 + -4 = 2.5 n = 2.5 Simplifying n = 2.5

Subproblem 2

n + 4 = -6.5 Simplifying n + 4 = -6.5 Reorder the terms: 4 + n = -6.5 Solving 4 + n = -6.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = -6.5 + -4 Combine like terms: 4 + -4 = 0 0 + n = -6.5 + -4 n = -6.5 + -4 Combine like terms: -6.5 + -4 = -10.5 n = -10.5 Simplifying n = -10.5

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.5, -10.5}

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