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Simplifying 3(5 + 2y) * 5y = 8 Reorder the terms for easier multiplication: 3 * 5y(5 + 2y) = 8 Multiply 3 * 5 15y(5 + 2y) = 8 (5 * 15y + 2y * 15y) = 8 (75y + 30y2) = 8 Solving 75y + 30y2 = 8 Solving for variable 'y'. Reorder the terms: -8 + 75y + 30y2 = 8 + -8 Combine like terms: 8 + -8 = 0 -8 + 75y + 30y2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -0.2666666667 + 2.5y + y2 = 0.0 Move the constant term to the right: Add '0.2666666667' to each side of the equation. -0.2666666667 + 2.5y + 0.2666666667 + y2 = 0.0 + 0.2666666667 Reorder the terms: -0.2666666667 + 0.2666666667 + 2.5y + y2 = 0.0 + 0.2666666667 Combine like terms: -0.2666666667 + 0.2666666667 = 0.0000000000 0.0000000000 + 2.5y + y2 = 0.0 + 0.2666666667 2.5y + y2 = 0.0 + 0.2666666667 Combine like terms: 0.0 + 0.2666666667 = 0.2666666667 2.5y + y2 = 0.2666666667 The y term is 2.5y. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5y + 1.5625 + y2 = 0.2666666667 + 1.5625 Reorder the terms: 1.5625 + 2.5y + y2 = 0.2666666667 + 1.5625 Combine like terms: 0.2666666667 + 1.5625 = 1.8291666667 1.5625 + 2.5y + y2 = 1.8291666667 Factor a perfect square on the left side: (y + 1.25)(y + 1.25) = 1.8291666667 Calculate the square root of the right side: 1.352466882 Break this problem into two subproblems by setting (y + 1.25) equal to 1.352466882 and -1.352466882.Subproblem 1
y + 1.25 = 1.352466882 Simplifying y + 1.25 = 1.352466882 Reorder the terms: 1.25 + y = 1.352466882 Solving 1.25 + y = 1.352466882 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + y = 1.352466882 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + y = 1.352466882 + -1.25 y = 1.352466882 + -1.25 Combine like terms: 1.352466882 + -1.25 = 0.102466882 y = 0.102466882 Simplifying y = 0.102466882Subproblem 2
y + 1.25 = -1.352466882 Simplifying y + 1.25 = -1.352466882 Reorder the terms: 1.25 + y = -1.352466882 Solving 1.25 + y = -1.352466882 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + y = -1.352466882 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + y = -1.352466882 + -1.25 y = -1.352466882 + -1.25 Combine like terms: -1.352466882 + -1.25 = -2.602466882 y = -2.602466882 Simplifying y = -2.602466882Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.102466882, -2.602466882}
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