5y^2+10y+4=2y^2+3y+1

Solution for 5y^2+10y+4=2y^2+3y+1 equation:

Simplifying
5y2 + 10y + 4 = 2y2 + 3y + 1

Reorder the terms:
4 + 10y + 5y2 = 2y2 + 3y + 1

Reorder the terms:
4 + 10y + 5y2 = 1 + 3y + 2y2

Solving
4 + 10y + 5y2 = 1 + 3y + 2y2

Solving for variable 'y'.

Reorder the terms:
4 + -1 + 10y + -3y + 5y2 + -2y2 = 1 + 3y + 2y2 + -1 + -3y + -2y2

Combine like terms: 4 + -1 = 3
3 + 10y + -3y + 5y2 + -2y2 = 1 + 3y + 2y2 + -1 + -3y + -2y2

Combine like terms: 10y + -3y = 7y
3 + 7y + 5y2 + -2y2 = 1 + 3y + 2y2 + -1 + -3y + -2y2

Combine like terms: 5y2 + -2y2 = 3y2
3 + 7y + 3y2 = 1 + 3y + 2y2 + -1 + -3y + -2y2

Reorder the terms:
3 + 7y + 3y2 = 1 + -1 + 3y + -3y + 2y2 + -2y2

Combine like terms: 1 + -1 = 0
3 + 7y + 3y2 = 0 + 3y + -3y + 2y2 + -2y2
3 + 7y + 3y2 = 3y + -3y + 2y2 + -2y2

Combine like terms: 3y + -3y = 0
3 + 7y + 3y2 = 0 + 2y2 + -2y2
3 + 7y + 3y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
3 + 7y + 3y2 = 0

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

Divide each side by '3'.
1 + 2.333333333y + y2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.333333333y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + 2.333333333y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.333333333y + y2 = 0 + -1
2.333333333y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.333333333y + y2 = -1

The y term is 2.333333333y.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333y + 1.361111112 + y2 = -1 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333y + y2 = -1 + 1.361111112

Combine like terms: -1 + 1.361111112 = 0.361111112
1.361111112 + 2.333333333y + y2 = 0.361111112

Factor a perfect square on the left side:
(y + 1.166666667)(y + 1.166666667) = 0.361111112

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(y + 1.166666667) equal to 0.600925213 and -0.600925213.

Subproblem 1
y + 1.166666667 = 0.600925213

Simplifying
y + 1.166666667 = 0.600925213

Reorder the terms:
1.166666667 + y = 0.600925213

Solving
1.166666667 + y = 0.600925213

Solving for variable 'y'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + y = 0.600925213 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + y = 0.600925213 + -1.166666667
y = 0.600925213 + -1.166666667

Combine like terms: 0.600925213 + -1.166666667 = -0.565741454
y = -0.565741454

Simplifying
y = -0.565741454

Subproblem 2
y + 1.166666667 = -0.600925213

Simplifying
y + 1.166666667 = -0.600925213

Reorder the terms:
1.166666667 + y = -0.600925213

Solving
1.166666667 + y = -0.600925213

Solving for variable 'y'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + y = -0.600925213 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + y = -0.600925213 + -1.166666667
y = -0.600925213 + -1.166666667

Combine like terms: -0.600925213 + -1.166666667 = -1.76759188
y = -1.76759188

Simplifying
y = -1.76759188

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.565741454, -1.76759188}