3x^2-105x+500=0

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Solution for 3x^2-105x+500=0 equation: 


Simplifying
3x2 + -105x + 500 = 0

Reorder the terms:
500 + -105x + 3x2 = 0

Solving
500 + -105x + 3x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
166.6666667 + -35x + x2 = 0

Move the constant term to the right:

Add '-166.6666667' to each side of the equation.
166.6666667 + -35x + -166.6666667 + x2 = 0 + -166.6666667

Reorder the terms:
166.6666667 + -166.6666667 + -35x + x2 = 0 + -166.6666667

Combine like terms: 166.6666667 + -166.6666667 = 0.0000000
0.0000000 + -35x + x2 = 0 + -166.6666667
-35x + x2 = 0 + -166.6666667

Combine like terms: 0 + -166.6666667 = -166.6666667
-35x + x2 = -166.6666667

The x term is -35x.  Take half its coefficient (-17.5).
Square it (306.25) and add it to both sides.

Add '306.25' to each side of the equation.
-35x + 306.25 + x2 = -166.6666667 + 306.25

Reorder the terms:
306.25 + -35x + x2 = -166.6666667 + 306.25

Combine like terms: -166.6666667 + 306.25 = 139.5833333
306.25 + -35x + x2 = 139.5833333

Factor a perfect square on the left side:
(x + -17.5)(x + -17.5) = 139.5833333

Calculate the square root of the right side: 11.814539064

Break this problem into two subproblems by setting 
(x + -17.5) equal to 11.814539064 and -11.814539064.


Subproblem 1
x + -17.5 = 11.814539064

Simplifying
x + -17.5 = 11.814539064

Reorder the terms:
-17.5 + x = 11.814539064

Solving
-17.5 + x = 11.814539064

Solving for variable 'x'.

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

Add '17.5' to each side of the equation.
-17.5 + 17.5 + x = 11.814539064 + 17.5

Combine like terms: -17.5 + 17.5 = 0.0
0.0 + x = 11.814539064 + 17.5
x = 11.814539064 + 17.5

Combine like terms: 11.814539064 + 17.5 = 29.314539064
x = 29.314539064

Simplifying
x = 29.314539064


Subproblem 2
x + -17.5 = -11.814539064

Simplifying
x + -17.5 = -11.814539064

Reorder the terms:
-17.5 + x = -11.814539064

Solving
-17.5 + x = -11.814539064

Solving for variable 'x'.

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

Add '17.5' to each side of the equation.
-17.5 + 17.5 + x = -11.814539064 + 17.5

Combine like terms: -17.5 + 17.5 = 0.0
0.0 + x = -11.814539064 + 17.5
x = -11.814539064 + 17.5

Combine like terms: -11.814539064 + 17.5 = 5.685460936
x = 5.685460936

Simplifying
x = 5.685460936


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {29.314539064, 5.685460936}

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