y^2+3y=8.5

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Solution for y^2+3y=8.5 equation: 


Simplifying
y2 + 3y = 8.5

Reorder the terms:
3y + y2 = 8.5

Solving
3y + y2 = 8.5

Solving for variable 'y'.

Reorder the terms:
-8.5 + 3y + y2 = 8.5 + -8.5

Combine like terms: 8.5 + -8.5 = 0.0
-8.5 + 3y + y2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '8.5' to each side of the equation.
-8.5 + 3y + 8.5 + y2 = 0.0 + 8.5

Reorder the terms:
-8.5 + 8.5 + 3y + y2 = 0.0 + 8.5

Combine like terms: -8.5 + 8.5 = 0.0
0.0 + 3y + y2 = 0.0 + 8.5
3y + y2 = 0.0 + 8.5

Combine like terms: 0.0 + 8.5 = 8.5
3y + y2 = 8.5

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 8.5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 8.5 + 2.25

Combine like terms: 8.5 + 2.25 = 10.75
2.25 + 3y + y2 = 10.75

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 10.75

Calculate the square root of the right side: 3.278719262

Break this problem into two subproblems by setting 
(y + 1.5) equal to 3.278719262 and -3.278719262.


Subproblem 1
y + 1.5 = 3.278719262

Simplifying
y + 1.5 = 3.278719262

Reorder the terms:
1.5 + y = 3.278719262

Solving
1.5 + y = 3.278719262

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 3.278719262 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 3.278719262 + -1.5
y = 3.278719262 + -1.5

Combine like terms: 3.278719262 + -1.5 = 1.778719262
y = 1.778719262

Simplifying
y = 1.778719262


Subproblem 2
y + 1.5 = -3.278719262

Simplifying
y + 1.5 = -3.278719262

Reorder the terms:
1.5 + y = -3.278719262

Solving
1.5 + y = -3.278719262

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -3.278719262 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -3.278719262 + -1.5
y = -3.278719262 + -1.5

Combine like terms: -3.278719262 + -1.5 = -4.778719262
y = -4.778719262

Simplifying
y = -4.778719262


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.778719262, -4.778719262}

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