2.25y^2+20y+31=0

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Solution for 2.25y^2+20y+31=0 equation: 


Simplifying
2.25y2 + 20y + 31 = 0

Reorder the terms:
31 + 20y + 2.25y2 = 0

Solving
31 + 20y + 2.25y2 = 0

Solving for variable 'y'.

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

Divide each side by '2.25'.
13.77777778 + 8.888888889y + y2 = 0

Move the constant term to the right:

Add '-13.77777778' to each side of the equation.
13.77777778 + 8.888888889y + -13.77777778 + y2 = 0 + -13.77777778

Reorder the terms:
13.77777778 + -13.77777778 + 8.888888889y + y2 = 0 + -13.77777778

Combine like terms: 13.77777778 + -13.77777778 = 0.00000000
0.00000000 + 8.888888889y + y2 = 0 + -13.77777778
8.888888889y + y2 = 0 + -13.77777778

Combine like terms: 0 + -13.77777778 = -13.77777778
8.888888889y + y2 = -13.77777778

The y term is 8.888888889y.  Take half its coefficient (4.444444445).
Square it (19.75308642) and add it to both sides.

Add '19.75308642' to each side of the equation.
8.888888889y + 19.75308642 + y2 = -13.77777778 + 19.75308642

Reorder the terms:
19.75308642 + 8.888888889y + y2 = -13.77777778 + 19.75308642

Combine like terms: -13.77777778 + 19.75308642 = 5.97530864
19.75308642 + 8.888888889y + y2 = 5.97530864

Factor a perfect square on the left side:
(y + 4.444444445)(y + 4.444444445) = 5.97530864

Calculate the square root of the right side: 2.444444444

Break this problem into two subproblems by setting 
(y + 4.444444445) equal to 2.444444444 and -2.444444444.


Subproblem 1
y + 4.444444445 = 2.444444444

Simplifying
y + 4.444444445 = 2.444444444

Reorder the terms:
4.444444445 + y = 2.444444444

Solving
4.444444445 + y = 2.444444444

Solving for variable 'y'.

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

Add '-4.444444445' to each side of the equation.
4.444444445 + -4.444444445 + y = 2.444444444 + -4.444444445

Combine like terms: 4.444444445 + -4.444444445 = 0.000000000
0.000000000 + y = 2.444444444 + -4.444444445
y = 2.444444444 + -4.444444445

Combine like terms: 2.444444444 + -4.444444445 = -2.000000001
y = -2.000000001

Simplifying
y = -2.000000001


Subproblem 2
y + 4.444444445 = -2.444444444

Simplifying
y + 4.444444445 = -2.444444444

Reorder the terms:
4.444444445 + y = -2.444444444

Solving
4.444444445 + y = -2.444444444

Solving for variable 'y'.

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

Add '-4.444444445' to each side of the equation.
4.444444445 + -4.444444445 + y = -2.444444444 + -4.444444445

Combine like terms: 4.444444445 + -4.444444445 = 0.000000000
0.000000000 + y = -2.444444444 + -4.444444445
y = -2.444444444 + -4.444444445

Combine like terms: -2.444444444 + -4.444444445 = -6.888888889
y = -6.888888889

Simplifying
y = -6.888888889


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-2.000000001, -6.888888889}

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