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Simplifying 16y + -1[3y(6 + -9y)] = 30y + [-1(3y + 2) + -1(y + 3)] 16y + -1[(6 * 3y + -9y * 3y)] = 30y + [-1(3y + 2) + -1(y + 3)] 16y + -1[(18y + -27y2)] = 30y + [-1(3y + 2) + -1(y + 3)] 16y + [18y * -1 + -27y2 * -1] = 30y + [-1(3y + 2) + -1(y + 3)] 16y + [-18y + 27y2] = 30y + [-1(3y + 2) + -1(y + 3)] Combine like terms: 16y + -18y = -2y -2y + 27y2 = 30y + [-1(3y + 2) + -1(y + 3)] Reorder the terms: -2y + 27y2 = 30y + [-1(2 + 3y) + -1(y + 3)] -2y + 27y2 = 30y + [(2 * -1 + 3y * -1) + -1(y + 3)] -2y + 27y2 = 30y + [(-2 + -3y) + -1(y + 3)] Reorder the terms: -2y + 27y2 = 30y + [-2 + -3y + -1(3 + y)] -2y + 27y2 = 30y + [-2 + -3y + (3 * -1 + y * -1)] -2y + 27y2 = 30y + [-2 + -3y + (-3 + -1y)] Reorder the terms: -2y + 27y2 = 30y + [-2 + -3 + -3y + -1y] Combine like terms: -2 + -3 = -5 -2y + 27y2 = 30y + [-5 + -3y + -1y] Combine like terms: -3y + -1y = -4y -2y + 27y2 = 30y + [-5 + -4y] Remove brackets around [-5 + -4y] -2y + 27y2 = 30y + -5 + -4y Reorder the terms: -2y + 27y2 = -5 + 30y + -4y Combine like terms: 30y + -4y = 26y -2y + 27y2 = -5 + 26y Solving -2y + 27y2 = -5 + 26y Solving for variable 'y'. Reorder the terms: 5 + -2y + -26y + 27y2 = -5 + 26y + 5 + -26y Combine like terms: -2y + -26y = -28y 5 + -28y + 27y2 = -5 + 26y + 5 + -26y Reorder the terms: 5 + -28y + 27y2 = -5 + 5 + 26y + -26y Combine like terms: -5 + 5 = 0 5 + -28y + 27y2 = 0 + 26y + -26y 5 + -28y + 27y2 = 26y + -26y Combine like terms: 26y + -26y = 0 5 + -28y + 27y2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. 0.1851851852 + -1.037037037y + y2 = 0 Move the constant term to the right: Add '-0.1851851852' to each side of the equation. 0.1851851852 + -1.037037037y + -0.1851851852 + y2 = 0 + -0.1851851852 Reorder the terms: 0.1851851852 + -0.1851851852 + -1.037037037y + y2 = 0 + -0.1851851852 Combine like terms: 0.1851851852 + -0.1851851852 = 0.0000000000 0.0000000000 + -1.037037037y + y2 = 0 + -0.1851851852 -1.037037037y + y2 = 0 + -0.1851851852 Combine like terms: 0 + -0.1851851852 = -0.1851851852 -1.037037037y + y2 = -0.1851851852 The y term is -1.037037037y. Take half its coefficient (-0.5185185185). Square it (0.2688614540) and add it to both sides. Add '0.2688614540' to each side of the equation. -1.037037037y + 0.2688614540 + y2 = -0.1851851852 + 0.2688614540 Reorder the terms: 0.2688614540 + -1.037037037y + y2 = -0.1851851852 + 0.2688614540 Combine like terms: -0.1851851852 + 0.2688614540 = 0.0836762688 0.2688614540 + -1.037037037y + y2 = 0.0836762688 Factor a perfect square on the left side: (y + -0.5185185185)(y + -0.5185185185) = 0.0836762688 Calculate the square root of the right side: 0.289268506 Break this problem into two subproblems by setting (y + -0.5185185185) equal to 0.289268506 and -0.289268506.Subproblem 1
y + -0.5185185185 = 0.289268506 Simplifying y + -0.5185185185 = 0.289268506 Reorder the terms: -0.5185185185 + y = 0.289268506 Solving -0.5185185185 + y = 0.289268506 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5185185185' to each side of the equation. -0.5185185185 + 0.5185185185 + y = 0.289268506 + 0.5185185185 Combine like terms: -0.5185185185 + 0.5185185185 = 0.0000000000 0.0000000000 + y = 0.289268506 + 0.5185185185 y = 0.289268506 + 0.5185185185 Combine like terms: 0.289268506 + 0.5185185185 = 0.8077870245 y = 0.8077870245 Simplifying y = 0.8077870245Subproblem 2
y + -0.5185185185 = -0.289268506 Simplifying y + -0.5185185185 = -0.289268506 Reorder the terms: -0.5185185185 + y = -0.289268506 Solving -0.5185185185 + y = -0.289268506 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5185185185' to each side of the equation. -0.5185185185 + 0.5185185185 + y = -0.289268506 + 0.5185185185 Combine like terms: -0.5185185185 + 0.5185185185 = 0.0000000000 0.0000000000 + y = -0.289268506 + 0.5185185185 y = -0.289268506 + 0.5185185185 Combine like terms: -0.289268506 + 0.5185185185 = 0.2292500125 y = 0.2292500125 Simplifying y = 0.2292500125Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.8077870245, 0.2292500125}
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