2y^2+9y=-6

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


Simplifying
2y2 + 9y = -6

Reorder the terms:
9y + 2y2 = -6

Solving
9y + 2y2 = -6

Solving for variable 'y'.

Reorder the terms:
6 + 9y + 2y2 = -6 + 6

Combine like terms: -6 + 6 = 0
6 + 9y + 2y2 = 0

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

Divide each side by '2'.
3 + 4.5y + y2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 4.5y + -3 + y2 = 0 + -3

Reorder the terms:
3 + -3 + 4.5y + y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 4.5y + y2 = 0 + -3
4.5y + y2 = 0 + -3

Combine like terms: 0 + -3 = -3
4.5y + y2 = -3

The y term is 4.5y.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5y + 5.0625 + y2 = -3 + 5.0625

Reorder the terms:
5.0625 + 4.5y + y2 = -3 + 5.0625

Combine like terms: -3 + 5.0625 = 2.0625
5.0625 + 4.5y + y2 = 2.0625

Factor a perfect square on the left side:
(y + 2.25)(y + 2.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(y + 2.25) equal to 1.436140662 and -1.436140662.

Subproblem 1

y + 2.25 = 1.436140662 Simplifying y + 2.25 = 1.436140662 Reorder the terms: 2.25 + y = 1.436140662 Solving 2.25 + y = 1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + y = 1.436140662 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + y = 1.436140662 + -2.25 y = 1.436140662 + -2.25 Combine like terms: 1.436140662 + -2.25 = -0.813859338 y = -0.813859338 Simplifying y = -0.813859338

Subproblem 2

y + 2.25 = -1.436140662 Simplifying y + 2.25 = -1.436140662 Reorder the terms: 2.25 + y = -1.436140662 Solving 2.25 + y = -1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + y = -1.436140662 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + y = -1.436140662 + -2.25 y = -1.436140662 + -2.25 Combine like terms: -1.436140662 + -2.25 = -3.686140662 y = -3.686140662 Simplifying y = -3.686140662

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.813859338, -3.686140662}

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