y+y^2=6.5

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Solution for y+y^2=6.5 equation: 


Simplifying
y + y2 = 6.5

Solving
y + y2 = 6.5

Solving for variable 'y'.

Reorder the terms:
-6.5 + y + y2 = 6.5 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
-6.5 + y + y2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + y + 6.5 + y2 = 0.0 + 6.5

Reorder the terms:
-6.5 + 6.5 + y + y2 = 0.0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + y + y2 = 0.0 + 6.5
y + y2 = 0.0 + 6.5

Combine like terms: 0.0 + 6.5 = 6.5
y + y2 = 6.5

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
y + 0.25 + y2 = 6.5 + 0.25

Reorder the terms:
0.25 + y + y2 = 6.5 + 0.25

Combine like terms: 6.5 + 0.25 = 6.75
0.25 + y + y2 = 6.75

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.598076211 and -2.598076211.


Subproblem 1
y + 0.5 = 2.598076211

Simplifying
y + 0.5 = 2.598076211

Reorder the terms:
0.5 + y = 2.598076211

Solving
0.5 + y = 2.598076211

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 2.598076211 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 2.598076211 + -0.5
y = 2.598076211 + -0.5

Combine like terms: 2.598076211 + -0.5 = 2.098076211
y = 2.098076211

Simplifying
y = 2.098076211


Subproblem 2
y + 0.5 = -2.598076211

Simplifying
y + 0.5 = -2.598076211

Reorder the terms:
0.5 + y = -2.598076211

Solving
0.5 + y = -2.598076211

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -2.598076211 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -2.598076211 + -0.5
y = -2.598076211 + -0.5

Combine like terms: -2.598076211 + -0.5 = -3.098076211
y = -3.098076211

Simplifying
y = -3.098076211


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.098076211, -3.098076211}

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