y^2+6y=9

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Solution for y^2+6y=9 equation:


Simplifying
y2 + 6y = 9

Reorder the terms:
6y + y2 = 9

Solving
6y + y2 = 9

Solving for variable 'y'.

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

Combine like terms: 9 + -9 = 0
-9 + 6y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 6y + 9 + y2 = 0 + 9

Reorder the terms:
-9 + 9 + 6y + y2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 6y + y2 = 0 + 9
6y + y2 = 0 + 9

Combine like terms: 0 + 9 = 9
6y + y2 = 9

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 = 9 + 9

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

Combine like terms: 9 + 9 = 18
9 + 6y + y2 = 18

Factor a perfect square on the left side:
(y + 3)(y + 3) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(y + 3) equal to 4.242640687 and -4.242640687.

Subproblem 1

y + 3 = 4.242640687 Simplifying y + 3 = 4.242640687 Reorder the terms: 3 + y = 4.242640687 Solving 3 + y = 4.242640687 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 = 4.242640687 + -3 Combine like terms: 3 + -3 = 0 0 + y = 4.242640687 + -3 y = 4.242640687 + -3 Combine like terms: 4.242640687 + -3 = 1.242640687 y = 1.242640687 Simplifying y = 1.242640687

Subproblem 2

y + 3 = -4.242640687 Simplifying y + 3 = -4.242640687 Reorder the terms: 3 + y = -4.242640687 Solving 3 + y = -4.242640687 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 = -4.242640687 + -3 Combine like terms: 3 + -3 = 0 0 + y = -4.242640687 + -3 y = -4.242640687 + -3 Combine like terms: -4.242640687 + -3 = -7.242640687 y = -7.242640687 Simplifying y = -7.242640687

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.242640687, -7.242640687}

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