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Simplifying 3y2 + 1 = -6y Reorder the terms: 1 + 3y2 = -6y Solving 1 + 3y2 = -6y Solving for variable 'y'. Reorder the terms: 1 + 6y + 3y2 = -6y + 6y Combine like terms: -6y + 6y = 0 1 + 6y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + 2y + y2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 2y + -0.3333333333 + y2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 2y + y2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 2y + y2 = 0 + -0.3333333333 2y + y2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 2y + y2 = -0.3333333333 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = -0.3333333333 + 1 Reorder the terms: 1 + 2y + y2 = -0.3333333333 + 1 Combine like terms: -0.3333333333 + 1 = 0.6666666667 1 + 2y + y2 = 0.6666666667 Factor a perfect square on the left side: (y + 1)(y + 1) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (y + 1) equal to 0.816496581 and -0.816496581.Subproblem 1
y + 1 = 0.816496581 Simplifying y + 1 = 0.816496581 Reorder the terms: 1 + y = 0.816496581 Solving 1 + y = 0.816496581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + y = 0.816496581 + -1 y = 0.816496581 + -1 Combine like terms: 0.816496581 + -1 = -0.183503419 y = -0.183503419 Simplifying y = -0.183503419Subproblem 2
y + 1 = -0.816496581 Simplifying y + 1 = -0.816496581 Reorder the terms: 1 + y = -0.816496581 Solving 1 + y = -0.816496581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + y = -0.816496581 + -1 y = -0.816496581 + -1 Combine like terms: -0.816496581 + -1 = -1.816496581 y = -1.816496581 Simplifying y = -1.816496581Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.183503419, -1.816496581}
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