10y^2-31y=15

Solution for 10y^2-31y=15 equation:

Simplifying
10y2 + -31y = 15

Reorder the terms:
-31y + 10y2 = 15

Solving
-31y + 10y2 = 15

Solving for variable 'y'.

Reorder the terms:
-15 + -31y + 10y2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + -31y + 10y2 = 0

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

Divide each side by '10'.
-1.5 + -3.1y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -3.1y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -3.1y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -3.1y + y2 = 0 + 1.5
-3.1y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-3.1y + y2 = 1.5

The y term is -3.1y.  Take half its coefficient (-1.55).
Square it (2.4025) and add it to both sides.

Add '2.4025' to each side of the equation.
-3.1y + 2.4025 + y2 = 1.5 + 2.4025

Reorder the terms:
2.4025 + -3.1y + y2 = 1.5 + 2.4025

Combine like terms: 1.5 + 2.4025 = 3.9025
2.4025 + -3.1y + y2 = 3.9025

Factor a perfect square on the left side:
(y + -1.55)(y + -1.55) = 3.9025

Calculate the square root of the right side: 1.975474627

Break this problem into two subproblems by setting 
(y + -1.55) equal to 1.975474627 and -1.975474627.

Subproblem 1
y + -1.55 = 1.975474627

Simplifying
y + -1.55 = 1.975474627

Reorder the terms:
-1.55 + y = 1.975474627

Solving
-1.55 + y = 1.975474627

Solving for variable 'y'.

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

Add '1.55' to each side of the equation.
-1.55 + 1.55 + y = 1.975474627 + 1.55

Combine like terms: -1.55 + 1.55 = 0.00
0.00 + y = 1.975474627 + 1.55
y = 1.975474627 + 1.55

Combine like terms: 1.975474627 + 1.55 = 3.525474627
y = 3.525474627

Simplifying
y = 3.525474627

Subproblem 2
y + -1.55 = -1.975474627

Simplifying
y + -1.55 = -1.975474627

Reorder the terms:
-1.55 + y = -1.975474627

Solving
-1.55 + y = -1.975474627

Solving for variable 'y'.

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

Add '1.55' to each side of the equation.
-1.55 + 1.55 + y = -1.975474627 + 1.55

Combine like terms: -1.55 + 1.55 = 0.00
0.00 + y = -1.975474627 + 1.55
y = -1.975474627 + 1.55

Combine like terms: -1.975474627 + 1.55 = -0.425474627
y = -0.425474627

Simplifying
y = -0.425474627

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.525474627, -0.425474627}