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Simplifying 2y(y + -0.5) = 23.5 Reorder the terms: 2y(-0.5 + y) = 23.5 (-0.5 * 2y + y * 2y) = 23.5 (-1y + 2y2) = 23.5 Solving -1y + 2y2 = 23.5 Solving for variable 'y'. Reorder the terms: -23.5 + -1y + 2y2 = 23.5 + -23.5 Combine like terms: 23.5 + -23.5 = 0.0 -23.5 + -1y + 2y2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11.75 + -0.5y + y2 = 0.0 Move the constant term to the right: Add '11.75' to each side of the equation. -11.75 + -0.5y + 11.75 + y2 = 0.0 + 11.75 Reorder the terms: -11.75 + 11.75 + -0.5y + y2 = 0.0 + 11.75 Combine like terms: -11.75 + 11.75 = 0.00 0.00 + -0.5y + y2 = 0.0 + 11.75 -0.5y + y2 = 0.0 + 11.75 Combine like terms: 0.0 + 11.75 = 11.75 -0.5y + y2 = 11.75 The y term is -0.5y. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5y + 0.0625 + y2 = 11.75 + 0.0625 Reorder the terms: 0.0625 + -0.5y + y2 = 11.75 + 0.0625 Combine like terms: 11.75 + 0.0625 = 11.8125 0.0625 + -0.5y + y2 = 11.8125 Factor a perfect square on the left side: (y + -0.25)(y + -0.25) = 11.8125 Calculate the square root of the right side: 3.436931771 Break this problem into two subproblems by setting (y + -0.25) equal to 3.436931771 and -3.436931771.Subproblem 1
y + -0.25 = 3.436931771 Simplifying y + -0.25 = 3.436931771 Reorder the terms: -0.25 + y = 3.436931771 Solving -0.25 + y = 3.436931771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = 3.436931771 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 3.436931771 + 0.25 y = 3.436931771 + 0.25 Combine like terms: 3.436931771 + 0.25 = 3.686931771 y = 3.686931771 Simplifying y = 3.686931771Subproblem 2
y + -0.25 = -3.436931771 Simplifying y + -0.25 = -3.436931771 Reorder the terms: -0.25 + y = -3.436931771 Solving -0.25 + y = -3.436931771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = -3.436931771 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -3.436931771 + 0.25 y = -3.436931771 + 0.25 Combine like terms: -3.436931771 + 0.25 = -3.186931771 y = -3.186931771 Simplifying y = -3.186931771Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.686931771, -3.186931771}
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