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Simplifying 3y(3y + 6) = 5y + 11 Reorder the terms: 3y(6 + 3y) = 5y + 11 (6 * 3y + 3y * 3y) = 5y + 11 (18y + 9y2) = 5y + 11 Reorder the terms: 18y + 9y2 = 11 + 5y Solving 18y + 9y2 = 11 + 5y Solving for variable 'y'. Reorder the terms: -11 + 18y + -5y + 9y2 = 11 + 5y + -11 + -5y Combine like terms: 18y + -5y = 13y -11 + 13y + 9y2 = 11 + 5y + -11 + -5y Reorder the terms: -11 + 13y + 9y2 = 11 + -11 + 5y + -5y Combine like terms: 11 + -11 = 0 -11 + 13y + 9y2 = 0 + 5y + -5y -11 + 13y + 9y2 = 5y + -5y Combine like terms: 5y + -5y = 0 -11 + 13y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.222222222 + 1.444444444y + y2 = 0 Move the constant term to the right: Add '1.222222222' to each side of the equation. -1.222222222 + 1.444444444y + 1.222222222 + y2 = 0 + 1.222222222 Reorder the terms: -1.222222222 + 1.222222222 + 1.444444444y + y2 = 0 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + 1.444444444y + y2 = 0 + 1.222222222 1.444444444y + y2 = 0 + 1.222222222 Combine like terms: 0 + 1.222222222 = 1.222222222 1.444444444y + y2 = 1.222222222 The y term is 1.444444444y. Take half its coefficient (0.722222222). Square it (0.5216049380) and add it to both sides. Add '0.5216049380' to each side of the equation. 1.444444444y + 0.5216049380 + y2 = 1.222222222 + 0.5216049380 Reorder the terms: 0.5216049380 + 1.444444444y + y2 = 1.222222222 + 0.5216049380 Combine like terms: 1.222222222 + 0.5216049380 = 1.74382716 0.5216049380 + 1.444444444y + y2 = 1.74382716 Factor a perfect square on the left side: (y + 0.722222222)(y + 0.722222222) = 1.74382716 Calculate the square root of the right side: 1.32054048 Break this problem into two subproblems by setting (y + 0.722222222) equal to 1.32054048 and -1.32054048.Subproblem 1
y + 0.722222222 = 1.32054048 Simplifying y + 0.722222222 = 1.32054048 Reorder the terms: 0.722222222 + y = 1.32054048 Solving 0.722222222 + y = 1.32054048 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.722222222' to each side of the equation. 0.722222222 + -0.722222222 + y = 1.32054048 + -0.722222222 Combine like terms: 0.722222222 + -0.722222222 = 0.000000000 0.000000000 + y = 1.32054048 + -0.722222222 y = 1.32054048 + -0.722222222 Combine like terms: 1.32054048 + -0.722222222 = 0.598318258 y = 0.598318258 Simplifying y = 0.598318258Subproblem 2
y + 0.722222222 = -1.32054048 Simplifying y + 0.722222222 = -1.32054048 Reorder the terms: 0.722222222 + y = -1.32054048 Solving 0.722222222 + y = -1.32054048 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.722222222' to each side of the equation. 0.722222222 + -0.722222222 + y = -1.32054048 + -0.722222222 Combine like terms: 0.722222222 + -0.722222222 = 0.000000000 0.000000000 + y = -1.32054048 + -0.722222222 y = -1.32054048 + -0.722222222 Combine like terms: -1.32054048 + -0.722222222 = -2.042762702 y = -2.042762702 Simplifying y = -2.042762702Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.598318258, -2.042762702}
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