n^2-34n-251=0

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Solution for n^2-34n-251=0 equation: 


Simplifying
n2 + -34n + -251 = 0

Reorder the terms:
-251 + -34n + n2 = 0

Solving
-251 + -34n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '251' to each side of the equation.
-251 + -34n + 251 + n2 = 0 + 251

Reorder the terms:
-251 + 251 + -34n + n2 = 0 + 251

Combine like terms: -251 + 251 = 0
0 + -34n + n2 = 0 + 251
-34n + n2 = 0 + 251

Combine like terms: 0 + 251 = 251
-34n + n2 = 251

The n term is -34n.  Take half its coefficient (-17).
Square it (289) and add it to both sides.

Add '289' to each side of the equation.
-34n + 289 + n2 = 251 + 289

Reorder the terms:
289 + -34n + n2 = 251 + 289

Combine like terms: 251 + 289 = 540
289 + -34n + n2 = 540

Factor a perfect square on the left side:
(n + -17)(n + -17) = 540

Calculate the square root of the right side: 23.237900077

Break this problem into two subproblems by setting 
(n + -17) equal to 23.237900077 and -23.237900077.


Subproblem 1
n + -17 = 23.237900077

Simplifying
n + -17 = 23.237900077

Reorder the terms:
-17 + n = 23.237900077

Solving
-17 + n = 23.237900077

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '17' to each side of the equation.
-17 + 17 + n = 23.237900077 + 17

Combine like terms: -17 + 17 = 0
0 + n = 23.237900077 + 17
n = 23.237900077 + 17

Combine like terms: 23.237900077 + 17 = 40.237900077
n = 40.237900077

Simplifying
n = 40.237900077


Subproblem 2
n + -17 = -23.237900077

Simplifying
n + -17 = -23.237900077

Reorder the terms:
-17 + n = -23.237900077

Solving
-17 + n = -23.237900077

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '17' to each side of the equation.
-17 + 17 + n = -23.237900077 + 17

Combine like terms: -17 + 17 = 0
0 + n = -23.237900077 + 17
n = -23.237900077 + 17

Combine like terms: -23.237900077 + 17 = -6.237900077
n = -6.237900077

Simplifying
n = -6.237900077


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {40.237900077, -6.237900077}

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