n^2-49+3n-441=0

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Solution for n^2-49+3n-441=0 equation:


Simplifying
n2 + -49 + 3n + -441 = 0

Reorder the terms:
-49 + -441 + 3n + n2 = 0

Combine like terms: -49 + -441 = -490
-490 + 3n + n2 = 0

Solving
-490 + 3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '490' to each side of the equation.
-490 + 3n + 490 + n2 = 0 + 490

Reorder the terms:
-490 + 490 + 3n + n2 = 0 + 490

Combine like terms: -490 + 490 = 0
0 + 3n + n2 = 0 + 490
3n + n2 = 0 + 490

Combine like terms: 0 + 490 = 490
3n + n2 = 490

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 490 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 490 + 2.25

Combine like terms: 490 + 2.25 = 492.25
2.25 + 3n + n2 = 492.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 492.25

Calculate the square root of the right side: 22.186707732

Break this problem into two subproblems by setting 
(n + 1.5) equal to 22.186707732 and -22.186707732.

Subproblem 1

n + 1.5 = 22.186707732 Simplifying n + 1.5 = 22.186707732 Reorder the terms: 1.5 + n = 22.186707732 Solving 1.5 + n = 22.186707732 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 22.186707732 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 22.186707732 + -1.5 n = 22.186707732 + -1.5 Combine like terms: 22.186707732 + -1.5 = 20.686707732 n = 20.686707732 Simplifying n = 20.686707732

Subproblem 2

n + 1.5 = -22.186707732 Simplifying n + 1.5 = -22.186707732 Reorder the terms: 1.5 + n = -22.186707732 Solving 1.5 + n = -22.186707732 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -22.186707732 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -22.186707732 + -1.5 n = -22.186707732 + -1.5 Combine like terms: -22.186707732 + -1.5 = -23.686707732 n = -23.686707732 Simplifying n = -23.686707732

Solution

The solution to the problem is based on the solutions from the subproblems. n = {20.686707732, -23.686707732}

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