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Simplifying 3 = 6y + -2y2 Solving 3 = 6y + -2y2 Solving for variable 'y'. Reorder the terms: 3 + -6y + 2y2 = 6y + -6y + -2y2 + 2y2 Combine like terms: 6y + -6y = 0 3 + -6y + 2y2 = 0 + -2y2 + 2y2 3 + -6y + 2y2 = -2y2 + 2y2 Combine like terms: -2y2 + 2y2 = 0 3 + -6y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3y + y2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3y + -1.5 + y2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3y + y2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3y + y2 = 0 + -1.5 -3y + y2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3y + y2 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + -3y + y2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3y + y2 = 0.75 Factor a perfect square on the left side: (y + -1.5)(y + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (y + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
y + -1.5 = 0.866025404 Simplifying y + -1.5 = 0.866025404 Reorder the terms: -1.5 + y = 0.866025404 Solving -1.5 + y = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 0.866025404 + 1.5 y = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 y = 2.366025404 Simplifying y = 2.366025404Subproblem 2
y + -1.5 = -0.866025404 Simplifying y + -1.5 = -0.866025404 Reorder the terms: -1.5 + y = -0.866025404 Solving -1.5 + y = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -0.866025404 + 1.5 y = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 y = 0.633974596 Simplifying y = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.366025404, 0.633974596}
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