5n+n^2-420=0

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Solution for 5n+n^2-420=0 equation:


Simplifying
5n + n2 + -420 = 0

Reorder the terms:
-420 + 5n + n2 = 0

Solving
-420 + 5n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '420' to each side of the equation.
-420 + 5n + 420 + n2 = 0 + 420

Reorder the terms:
-420 + 420 + 5n + n2 = 0 + 420

Combine like terms: -420 + 420 = 0
0 + 5n + n2 = 0 + 420
5n + n2 = 0 + 420

Combine like terms: 0 + 420 = 420
5n + n2 = 420

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 420 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 420 + 6.25

Combine like terms: 420 + 6.25 = 426.25
6.25 + 5n + n2 = 426.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 426.25

Calculate the square root of the right side: 20.645822822

Break this problem into two subproblems by setting 
(n + 2.5) equal to 20.645822822 and -20.645822822.

Subproblem 1

n + 2.5 = 20.645822822 Simplifying n + 2.5 = 20.645822822 Reorder the terms: 2.5 + n = 20.645822822 Solving 2.5 + n = 20.645822822 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 20.645822822 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 20.645822822 + -2.5 n = 20.645822822 + -2.5 Combine like terms: 20.645822822 + -2.5 = 18.145822822 n = 18.145822822 Simplifying n = 18.145822822

Subproblem 2

n + 2.5 = -20.645822822 Simplifying n + 2.5 = -20.645822822 Reorder the terms: 2.5 + n = -20.645822822 Solving 2.5 + n = -20.645822822 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -20.645822822 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -20.645822822 + -2.5 n = -20.645822822 + -2.5 Combine like terms: -20.645822822 + -2.5 = -23.145822822 n = -23.145822822 Simplifying n = -23.145822822

Solution

The solution to the problem is based on the solutions from the subproblems. n = {18.145822822, -23.145822822}

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