n^2+39n-420=0

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Solution for n^2+39n-420=0 equation:


Simplifying
n2 + 39n + -420 = 0

Reorder the terms:
-420 + 39n + n2 = 0

Solving
-420 + 39n + 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 + 39n + 420 + n2 = 0 + 420

Reorder the terms:
-420 + 420 + 39n + n2 = 0 + 420

Combine like terms: -420 + 420 = 0
0 + 39n + n2 = 0 + 420
39n + n2 = 0 + 420

Combine like terms: 0 + 420 = 420
39n + n2 = 420

The n term is 39n.  Take half its coefficient (19.5).
Square it (380.25) and add it to both sides.

Add '380.25' to each side of the equation.
39n + 380.25 + n2 = 420 + 380.25

Reorder the terms:
380.25 + 39n + n2 = 420 + 380.25

Combine like terms: 420 + 380.25 = 800.25
380.25 + 39n + n2 = 800.25

Factor a perfect square on the left side:
(n + 19.5)(n + 19.5) = 800.25

Calculate the square root of the right side: 28.28869032

Break this problem into two subproblems by setting 
(n + 19.5) equal to 28.28869032 and -28.28869032.

Subproblem 1

n + 19.5 = 28.28869032 Simplifying n + 19.5 = 28.28869032 Reorder the terms: 19.5 + n = 28.28869032 Solving 19.5 + n = 28.28869032 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-19.5' to each side of the equation. 19.5 + -19.5 + n = 28.28869032 + -19.5 Combine like terms: 19.5 + -19.5 = 0.0 0.0 + n = 28.28869032 + -19.5 n = 28.28869032 + -19.5 Combine like terms: 28.28869032 + -19.5 = 8.78869032 n = 8.78869032 Simplifying n = 8.78869032

Subproblem 2

n + 19.5 = -28.28869032 Simplifying n + 19.5 = -28.28869032 Reorder the terms: 19.5 + n = -28.28869032 Solving 19.5 + n = -28.28869032 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-19.5' to each side of the equation. 19.5 + -19.5 + n = -28.28869032 + -19.5 Combine like terms: 19.5 + -19.5 = 0.0 0.0 + n = -28.28869032 + -19.5 n = -28.28869032 + -19.5 Combine like terms: -28.28869032 + -19.5 = -47.78869032 n = -47.78869032 Simplifying n = -47.78869032

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.78869032, -47.78869032}

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