y=y^2+6y-1

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Solution for y=y^2+6y-1 equation: 


Simplifying
y = y2 + 6y + -1

Reorder the terms:
y = -1 + 6y + y2

Solving
y = -1 + 6y + y2

Solving for variable 'y'.

Reorder the terms:
1 + y + -6y + -1y2 = -1 + 6y + y2 + 1 + -6y + -1y2

Combine like terms: y + -6y = -5y
1 + -5y + -1y2 = -1 + 6y + y2 + 1 + -6y + -1y2

Reorder the terms:
1 + -5y + -1y2 = -1 + 1 + 6y + -6y + y2 + -1y2

Combine like terms: -1 + 1 = 0
1 + -5y + -1y2 = 0 + 6y + -6y + y2 + -1y2
1 + -5y + -1y2 = 6y + -6y + y2 + -1y2

Combine like terms: 6y + -6y = 0
1 + -5y + -1y2 = 0 + y2 + -1y2
1 + -5y + -1y2 = y2 + -1y2

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

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

Divide each side by '-1'.
-1 + 5y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 5y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 5y + y2 = 0 + 1

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

Combine like terms: 0 + 1 = 1
5y + y2 = 1

The y term is 5y.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5y + 6.25 + y2 = 1 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 1 + 6.25

Combine like terms: 1 + 6.25 = 7.25
6.25 + 5y + y2 = 7.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + 2.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
y + 2.5 = 2.692582404

Simplifying
y + 2.5 = 2.692582404

Reorder the terms:
2.5 + y = 2.692582404

Solving
2.5 + y = 2.692582404

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = 2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 2.692582404 + -2.5
y = 2.692582404 + -2.5

Combine like terms: 2.692582404 + -2.5 = 0.192582404
y = 0.192582404

Simplifying
y = 0.192582404


Subproblem 2
y + 2.5 = -2.692582404

Simplifying
y + 2.5 = -2.692582404

Reorder the terms:
2.5 + y = -2.692582404

Solving
2.5 + y = -2.692582404

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = -2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -2.692582404 + -2.5
y = -2.692582404 + -2.5

Combine like terms: -2.692582404 + -2.5 = -5.192582404
y = -5.192582404

Simplifying
y = -5.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.192582404, -5.192582404}

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