-y^2=6y-1

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


Simplifying
-1y2 = 6y + -1

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

Solving
-1y2 = -1 + 6y

Solving for variable 'y'.

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

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

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

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

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

Divide each side by '-1'.
-1 + 6y + y2 = 0

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 = 10
9 + 6y + y2 = 10

Factor a perfect square on the left side:
(y + 3)(y + 3) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(y + 3) equal to 3.16227766 and -3.16227766.

Subproblem 1

y + 3 = 3.16227766 Simplifying y + 3 = 3.16227766 Reorder the terms: 3 + y = 3.16227766 Solving 3 + y = 3.16227766 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 = 3.16227766 + -3 Combine like terms: 3 + -3 = 0 0 + y = 3.16227766 + -3 y = 3.16227766 + -3 Combine like terms: 3.16227766 + -3 = 0.16227766 y = 0.16227766 Simplifying y = 0.16227766

Subproblem 2

y + 3 = -3.16227766 Simplifying y + 3 = -3.16227766 Reorder the terms: 3 + y = -3.16227766 Solving 3 + y = -3.16227766 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 = -3.16227766 + -3 Combine like terms: 3 + -3 = 0 0 + y = -3.16227766 + -3 y = -3.16227766 + -3 Combine like terms: -3.16227766 + -3 = -6.16227766 y = -6.16227766 Simplifying y = -6.16227766

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.16227766, -6.16227766}

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