y^2+1=2y^2+4y-14

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Solution for y^2+1=2y^2+4y-14 equation: 


Simplifying
y2 + 1 = 2y2 + 4y + -14

Reorder the terms:
1 + y2 = 2y2 + 4y + -14

Reorder the terms:
1 + y2 = -14 + 4y + 2y2

Solving
1 + y2 = -14 + 4y + 2y2

Solving for variable 'y'.

Reorder the terms:
1 + 14 + -4y + y2 + -2y2 = -14 + 4y + 2y2 + 14 + -4y + -2y2

Combine like terms: 1 + 14 = 15
15 + -4y + y2 + -2y2 = -14 + 4y + 2y2 + 14 + -4y + -2y2

Combine like terms: y2 + -2y2 = -1y2
15 + -4y + -1y2 = -14 + 4y + 2y2 + 14 + -4y + -2y2

Reorder the terms:
15 + -4y + -1y2 = -14 + 14 + 4y + -4y + 2y2 + -2y2

Combine like terms: -14 + 14 = 0
15 + -4y + -1y2 = 0 + 4y + -4y + 2y2 + -2y2
15 + -4y + -1y2 = 4y + -4y + 2y2 + -2y2

Combine like terms: 4y + -4y = 0
15 + -4y + -1y2 = 0 + 2y2 + -2y2
15 + -4y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
15 + -4y + -1y2 = 0

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

Divide each side by '-1'.
-15 + 4y + y2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 4y + 15 + y2 = 0 + 15

Reorder the terms:
-15 + 15 + 4y + y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 4y + y2 = 0 + 15
4y + y2 = 0 + 15

Combine like terms: 0 + 15 = 15
4y + y2 = 15

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = 15 + 4

Reorder the terms:
4 + 4y + y2 = 15 + 4

Combine like terms: 15 + 4 = 19
4 + 4y + y2 = 19

Factor a perfect square on the left side:
(y + 2)(y + 2) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(y + 2) equal to 4.358898944 and -4.358898944.


Subproblem 1
y + 2 = 4.358898944

Simplifying
y + 2 = 4.358898944

Reorder the terms:
2 + y = 4.358898944

Solving
2 + y = 4.358898944

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = 4.358898944 + -2

Combine like terms: 2 + -2 = 0
0 + y = 4.358898944 + -2
y = 4.358898944 + -2

Combine like terms: 4.358898944 + -2 = 2.358898944
y = 2.358898944

Simplifying
y = 2.358898944


Subproblem 2
y + 2 = -4.358898944

Simplifying
y + 2 = -4.358898944

Reorder the terms:
2 + y = -4.358898944

Solving
2 + y = -4.358898944

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = -4.358898944 + -2

Combine like terms: 2 + -2 = 0
0 + y = -4.358898944 + -2
y = -4.358898944 + -2

Combine like terms: -4.358898944 + -2 = -6.358898944
y = -6.358898944

Simplifying
y = -6.358898944


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.358898944, -6.358898944}

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