10y^2+23y+5=0

Solution for 10y^2+23y+5=0 equation:

Simplifying
10y2 + 23y + 5 = 0

Reorder the terms:
5 + 23y + 10y2 = 0

Solving
5 + 23y + 10y2 = 0

Solving for variable 'y'.

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

Divide each side by '10'.
0.5 + 2.3y + y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.3y + -0.5 + y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.3y + y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.3y + y2 = 0 + -0.5
2.3y + y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.3y + y2 = -0.5

The y term is 2.3y.  Take half its coefficient (1.15).
Square it (1.3225) and add it to both sides.

Add '1.3225' to each side of the equation.
2.3y + 1.3225 + y2 = -0.5 + 1.3225

Reorder the terms:
1.3225 + 2.3y + y2 = -0.5 + 1.3225

Combine like terms: -0.5 + 1.3225 = 0.8225
1.3225 + 2.3y + y2 = 0.8225

Factor a perfect square on the left side:
(y + 1.15)(y + 1.15) = 0.8225

Calculate the square root of the right side: 0.906917857

Break this problem into two subproblems by setting 
(y + 1.15) equal to 0.906917857 and -0.906917857.

Subproblem 1
y + 1.15 = 0.906917857

Simplifying
y + 1.15 = 0.906917857

Reorder the terms:
1.15 + y = 0.906917857

Solving
1.15 + y = 0.906917857

Solving for variable 'y'.

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

Add '-1.15' to each side of the equation.
1.15 + -1.15 + y = 0.906917857 + -1.15

Combine like terms: 1.15 + -1.15 = 0.00
0.00 + y = 0.906917857 + -1.15
y = 0.906917857 + -1.15

Combine like terms: 0.906917857 + -1.15 = -0.243082143
y = -0.243082143

Simplifying
y = -0.243082143

Subproblem 2
y + 1.15 = -0.906917857

Simplifying
y + 1.15 = -0.906917857

Reorder the terms:
1.15 + y = -0.906917857

Solving
1.15 + y = -0.906917857

Solving for variable 'y'.

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

Add '-1.15' to each side of the equation.
1.15 + -1.15 + y = -0.906917857 + -1.15

Combine like terms: 1.15 + -1.15 = 0.00
0.00 + y = -0.906917857 + -1.15
y = -0.906917857 + -1.15

Combine like terms: -0.906917857 + -1.15 = -2.056917857
y = -2.056917857

Simplifying
y = -2.056917857

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.243082143, -2.056917857}