y^2+25=2y^2+9y+5

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Solution for y^2+25=2y^2+9y+5 equation: 


Simplifying
y2 + 25 = 2y2 + 9y + 5

Reorder the terms:
25 + y2 = 2y2 + 9y + 5

Reorder the terms:
25 + y2 = 5 + 9y + 2y2

Solving
25 + y2 = 5 + 9y + 2y2

Solving for variable 'y'.

Reorder the terms:
25 + -5 + -9y + y2 + -2y2 = 5 + 9y + 2y2 + -5 + -9y + -2y2

Combine like terms: 25 + -5 = 20
20 + -9y + y2 + -2y2 = 5 + 9y + 2y2 + -5 + -9y + -2y2

Combine like terms: y2 + -2y2 = -1y2
20 + -9y + -1y2 = 5 + 9y + 2y2 + -5 + -9y + -2y2

Reorder the terms:
20 + -9y + -1y2 = 5 + -5 + 9y + -9y + 2y2 + -2y2

Combine like terms: 5 + -5 = 0
20 + -9y + -1y2 = 0 + 9y + -9y + 2y2 + -2y2
20 + -9y + -1y2 = 9y + -9y + 2y2 + -2y2

Combine like terms: 9y + -9y = 0
20 + -9y + -1y2 = 0 + 2y2 + -2y2
20 + -9y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
20 + -9y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-20 + 9y + y2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 9y + 20 + y2 = 0 + 20

Reorder the terms:
-20 + 20 + 9y + y2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 9y + y2 = 0 + 20
9y + y2 = 0 + 20

Combine like terms: 0 + 20 = 20
9y + y2 = 20

The y term is 9y.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9y + 20.25 + y2 = 20 + 20.25

Reorder the terms:
20.25 + 9y + y2 = 20 + 20.25

Combine like terms: 20 + 20.25 = 40.25
20.25 + 9y + y2 = 40.25

Factor a perfect square on the left side:
(y + 4.5)(y + 4.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(y + 4.5) equal to 6.34428877 and -6.34428877.


Subproblem 1
y + 4.5 = 6.34428877

Simplifying
y + 4.5 = 6.34428877

Reorder the terms:
4.5 + y = 6.34428877

Solving
4.5 + y = 6.34428877

Solving for variable 'y'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + y = 6.34428877 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = 6.34428877 + -4.5
y = 6.34428877 + -4.5

Combine like terms: 6.34428877 + -4.5 = 1.84428877
y = 1.84428877

Simplifying
y = 1.84428877


Subproblem 2
y + 4.5 = -6.34428877

Simplifying
y + 4.5 = -6.34428877

Reorder the terms:
4.5 + y = -6.34428877

Solving
4.5 + y = -6.34428877

Solving for variable 'y'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + y = -6.34428877 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = -6.34428877 + -4.5
y = -6.34428877 + -4.5

Combine like terms: -6.34428877 + -4.5 = -10.84428877
y = -10.84428877

Simplifying
y = -10.84428877


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.84428877, -10.84428877}

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