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Simplifying 3y2 + 9y + 2 = 0 Reorder the terms: 2 + 9y + 3y2 = 0 Solving 2 + 9y + 3y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + 3y + y2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 3y + -0.6666666667 + y2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 3y + y2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 3y + y2 = 0 + -0.6666666667 3y + y2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 3y + y2 = -0.6666666667 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = -0.6666666667 + 2.25 Reorder the terms: 2.25 + 3y + y2 = -0.6666666667 + 2.25 Combine like terms: -0.6666666667 + 2.25 = 1.5833333333 2.25 + 3y + y2 = 1.5833333333 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 1.5833333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (y + 1.5) equal to 1.258305739 and -1.258305739.Subproblem 1
y + 1.5 = 1.258305739 Simplifying y + 1.5 = 1.258305739 Reorder the terms: 1.5 + y = 1.258305739 Solving 1.5 + y = 1.258305739 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.258305739 + -1.5 y = 1.258305739 + -1.5 Combine like terms: 1.258305739 + -1.5 = -0.241694261 y = -0.241694261 Simplifying y = -0.241694261Subproblem 2
y + 1.5 = -1.258305739 Simplifying y + 1.5 = -1.258305739 Reorder the terms: 1.5 + y = -1.258305739 Solving 1.5 + y = -1.258305739 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.258305739 + -1.5 y = -1.258305739 + -1.5 Combine like terms: -1.258305739 + -1.5 = -2.758305739 y = -2.758305739 Simplifying y = -2.758305739Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.241694261, -2.758305739}
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