y^2+6y+1=0

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Solution for y^2+6y+1=0 equation:


Simplifying
y2 + 6y + 1 = 0

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

Solving
1 + 6y + y2 = 0

Solving for variable 'y'.

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 = -0.171572875 y = -0.171572875 Simplifying y = -0.171572875

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 = -5.828427125 y = -5.828427125 Simplifying y = -5.828427125

Solution

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

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