2x^2-5x+1=3

Solution for 2x^2-5x+1=3 equation:

Simplifying
2x2 + -5x + 1 = 3

Reorder the terms:
1 + -5x + 2x2 = 3

Solving
1 + -5x + 2x2 = 3

Solving for variable 'x'.

Reorder the terms:
1 + -3 + -5x + 2x2 = 3 + -3

Combine like terms: 1 + -3 = -2
-2 + -5x + 2x2 = 3 + -3

Combine like terms: 3 + -3 = 0
-2 + -5x + 2x2 = 0

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

Divide each side by '2'.
-1 + -2.5x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.5x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.5x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.5x + x2 = 0 + 1
-2.5x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.5x + x2 = 1

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 1 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 1 + 1.5625

Combine like terms: 1 + 1.5625 = 2.5625
1.5625 + -2.5x + x2 = 2.5625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.600781059 and -1.600781059.

Subproblem 1
x + -1.25 = 1.600781059

Simplifying
x + -1.25 = 1.600781059

Reorder the terms:
-1.25 + x = 1.600781059

Solving
-1.25 + x = 1.600781059

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 1.600781059 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 1.600781059 + 1.25
x = 1.600781059 + 1.25

Combine like terms: 1.600781059 + 1.25 = 2.850781059
x = 2.850781059

Simplifying
x = 2.850781059

Subproblem 2
x + -1.25 = -1.600781059

Simplifying
x + -1.25 = -1.600781059

Reorder the terms:
-1.25 + x = -1.600781059

Solving
-1.25 + x = -1.600781059

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -1.600781059 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -1.600781059 + 1.25
x = -1.600781059 + 1.25

Combine like terms: -1.600781059 + 1.25 = -0.350781059
x = -0.350781059

Simplifying
x = -0.350781059

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.850781059, -0.350781059}