y^2=6y-1

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


Simplifying
y2 = 6y + -1

Reorder the terms:
y2 = -1 + 6y

Solving
y2 = -1 + 6y

Solving for variable 'y'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -6y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + -6y + y2 = 0 + -1

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

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

The y term is -6y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6y + 9 + y2 = -1 + 9

Reorder the terms:
9 + -6y + y2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + -6y + y2 = 8

Factor a perfect square on the left side:
(y + -3)(y + -3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(y + -3) equal to 2.828427125 and -2.828427125.

Subproblem 1

y + -3 = 2.828427125 Simplifying y + -3 = 2.828427125 Reorder the terms: -3 + y = 2.828427125 Solving -3 + y = 2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + y = 2.828427125 + 3 y = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 y = 5.828427125 Simplifying y = 5.828427125

Subproblem 2

y + -3 = -2.828427125 Simplifying y + -3 = -2.828427125 Reorder the terms: -3 + y = -2.828427125 Solving -3 + y = -2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + y = -2.828427125 + 3 y = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 y = 0.171572875 Simplifying y = 0.171572875

Solution

The solution to the problem is based on the solutions from the subproblems. y = {5.828427125, 0.171572875}

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