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Simplifying 2y2 + 11 = 10y Reorder the terms: 11 + 2y2 = 10y Solving 11 + 2y2 = 10y Solving for variable 'y'. Reorder the terms: 11 + -10y + 2y2 = 10y + -10y Combine like terms: 10y + -10y = 0 11 + -10y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 5.5 + -5y + y2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + -5y + -5.5 + y2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + -5y + y2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + -5y + y2 = 0 + -5.5 -5y + y2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 -5y + y2 = -5.5 The y term is -5y. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5y + 6.25 + y2 = -5.5 + 6.25 Reorder the terms: 6.25 + -5y + y2 = -5.5 + 6.25 Combine like terms: -5.5 + 6.25 = 0.75 6.25 + -5y + y2 = 0.75 Factor a perfect square on the left side: (y + -2.5)(y + -2.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (y + -2.5) equal to 0.866025404 and -0.866025404.Subproblem 1
y + -2.5 = 0.866025404 Simplifying y + -2.5 = 0.866025404 Reorder the terms: -2.5 + y = 0.866025404 Solving -2.5 + y = 0.866025404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = 0.866025404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = 0.866025404 + 2.5 y = 0.866025404 + 2.5 Combine like terms: 0.866025404 + 2.5 = 3.366025404 y = 3.366025404 Simplifying y = 3.366025404Subproblem 2
y + -2.5 = -0.866025404 Simplifying y + -2.5 = -0.866025404 Reorder the terms: -2.5 + y = -0.866025404 Solving -2.5 + y = -0.866025404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = -0.866025404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = -0.866025404 + 2.5 y = -0.866025404 + 2.5 Combine like terms: -0.866025404 + 2.5 = 1.633974596 y = 1.633974596 Simplifying y = 1.633974596Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.366025404, 1.633974596}
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