x^2-51x+1=0

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Solution for x^2-51x+1=0 equation: 


Simplifying
x2 + -51x + 1 = 0

Reorder the terms:
1 + -51x + x2 = 0

Solving
1 + -51x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -51x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -51x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -51x + x2 = 0 + -1
-51x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-51x + x2 = -1

The x term is -51x.  Take half its coefficient (-25.5).
Square it (650.25) and add it to both sides.

Add '650.25' to each side of the equation.
-51x + 650.25 + x2 = -1 + 650.25

Reorder the terms:
650.25 + -51x + x2 = -1 + 650.25

Combine like terms: -1 + 650.25 = 649.25
650.25 + -51x + x2 = 649.25

Factor a perfect square on the left side:
(x + -25.5)(x + -25.5) = 649.25

Calculate the square root of the right side: 25.480384612

Break this problem into two subproblems by setting 
(x + -25.5) equal to 25.480384612 and -25.480384612.


Subproblem 1
x + -25.5 = 25.480384612

Simplifying
x + -25.5 = 25.480384612

Reorder the terms:
-25.5 + x = 25.480384612

Solving
-25.5 + x = 25.480384612

Solving for variable 'x'.

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

Add '25.5' to each side of the equation.
-25.5 + 25.5 + x = 25.480384612 + 25.5

Combine like terms: -25.5 + 25.5 = 0.0
0.0 + x = 25.480384612 + 25.5
x = 25.480384612 + 25.5

Combine like terms: 25.480384612 + 25.5 = 50.980384612
x = 50.980384612

Simplifying
x = 50.980384612


Subproblem 2
x + -25.5 = -25.480384612

Simplifying
x + -25.5 = -25.480384612

Reorder the terms:
-25.5 + x = -25.480384612

Solving
-25.5 + x = -25.480384612

Solving for variable 'x'.

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

Add '25.5' to each side of the equation.
-25.5 + 25.5 + x = -25.480384612 + 25.5

Combine like terms: -25.5 + 25.5 = 0.0
0.0 + x = -25.480384612 + 25.5
x = -25.480384612 + 25.5

Combine like terms: -25.480384612 + 25.5 = 0.019615388
x = 0.019615388

Simplifying
x = 0.019615388


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {50.980384612, 0.019615388}

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