2y^2-3=5y+1

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

Simplifying
2y2 + -3 = 5y + 1

Reorder the terms:
-3 + 2y2 = 5y + 1

Reorder the terms:
-3 + 2y2 = 1 + 5y

Solving
-3 + 2y2 = 1 + 5y

Solving for variable 'y'.

Reorder the terms:
-3 + -1 + -5y + 2y2 = 1 + 5y + -1 + -5y

Combine like terms: -3 + -1 = -4
-4 + -5y + 2y2 = 1 + 5y + -1 + -5y

Reorder the terms:
-4 + -5y + 2y2 = 1 + -1 + 5y + -5y

Combine like terms: 1 + -1 = 0
-4 + -5y + 2y2 = 0 + 5y + -5y
-4 + -5y + 2y2 = 5y + -5y

Combine like terms: 5y + -5y = 0
-4 + -5y + 2y2 = 0

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

Divide each side by '2'.
-2 + -2.5y + y2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2.5y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + -2.5y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -2.5y + y2 = 0 + 2
-2.5y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
-2.5y + y2 = 2

The y term is -2.5y.  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.
-2.5y + 1.5625 + y2 = 2 + 1.5625

Reorder the terms:
1.5625 + -2.5y + y2 = 2 + 1.5625

Combine like terms: 2 + 1.5625 = 3.5625
1.5625 + -2.5y + y2 = 3.5625

Factor a perfect square on the left side:
(y + -1.25)(y + -1.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(y + -1.25) equal to 1.887458609 and -1.887458609.

Subproblem 1
y + -1.25 = 1.887458609

Simplifying
y + -1.25 = 1.887458609

Reorder the terms:
-1.25 + y = 1.887458609

Solving
-1.25 + y = 1.887458609

Solving for variable 'y'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + y = 1.887458609 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + y = 1.887458609 + 1.25
y = 1.887458609 + 1.25

Combine like terms: 1.887458609 + 1.25 = 3.137458609
y = 3.137458609

Simplifying
y = 3.137458609

Subproblem 2
y + -1.25 = -1.887458609

Simplifying
y + -1.25 = -1.887458609

Reorder the terms:
-1.25 + y = -1.887458609

Solving
-1.25 + y = -1.887458609

Solving for variable 'y'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + y = -1.887458609 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + y = -1.887458609 + 1.25
y = -1.887458609 + 1.25

Combine like terms: -1.887458609 + 1.25 = -0.637458609
y = -0.637458609

Simplifying
y = -0.637458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.137458609, -0.637458609}