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Simplifying 3(y + 6) * y = 2(y + 9) Reorder the terms: 3(6 + y) * y = 2(y + 9) Reorder the terms for easier multiplication: 3y(6 + y) = 2(y + 9) (6 * 3y + y * 3y) = 2(y + 9) (18y + 3y2) = 2(y + 9) Reorder the terms: 18y + 3y2 = 2(9 + y) 18y + 3y2 = (9 * 2 + y * 2) 18y + 3y2 = (18 + 2y) Solving 18y + 3y2 = 18 + 2y Solving for variable 'y'. Reorder the terms: -18 + 18y + -2y + 3y2 = 18 + 2y + -18 + -2y Combine like terms: 18y + -2y = 16y -18 + 16y + 3y2 = 18 + 2y + -18 + -2y Reorder the terms: -18 + 16y + 3y2 = 18 + -18 + 2y + -2y Combine like terms: 18 + -18 = 0 -18 + 16y + 3y2 = 0 + 2y + -2y -18 + 16y + 3y2 = 2y + -2y Combine like terms: 2y + -2y = 0 -18 + 16y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6 + 5.333333333y + y2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 5.333333333y + 6 + y2 = 0 + 6 Reorder the terms: -6 + 6 + 5.333333333y + y2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 5.333333333y + y2 = 0 + 6 5.333333333y + y2 = 0 + 6 Combine like terms: 0 + 6 = 6 5.333333333y + y2 = 6 The y term is 5.333333333y. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333y + 7.111111113 + y2 = 6 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333y + y2 = 6 + 7.111111113 Combine like terms: 6 + 7.111111113 = 13.111111113 7.111111113 + 5.333333333y + y2 = 13.111111113 Factor a perfect square on the left side: (y + 2.666666667)(y + 2.666666667) = 13.111111113 Calculate the square root of the right side: 3.620926831 Break this problem into two subproblems by setting (y + 2.666666667) equal to 3.620926831 and -3.620926831.Subproblem 1
y + 2.666666667 = 3.620926831 Simplifying y + 2.666666667 = 3.620926831 Reorder the terms: 2.666666667 + y = 3.620926831 Solving 2.666666667 + y = 3.620926831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + y = 3.620926831 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + y = 3.620926831 + -2.666666667 y = 3.620926831 + -2.666666667 Combine like terms: 3.620926831 + -2.666666667 = 0.954260164 y = 0.954260164 Simplifying y = 0.954260164Subproblem 2
y + 2.666666667 = -3.620926831 Simplifying y + 2.666666667 = -3.620926831 Reorder the terms: 2.666666667 + y = -3.620926831 Solving 2.666666667 + y = -3.620926831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + y = -3.620926831 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + y = -3.620926831 + -2.666666667 y = -3.620926831 + -2.666666667 Combine like terms: -3.620926831 + -2.666666667 = -6.287593498 y = -6.287593498 Simplifying y = -6.287593498Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.954260164, -6.287593498}
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