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Simplifying 5y + 3y(y + 2) = 54 Reorder the terms: 5y + 3y(2 + y) = 54 5y + (2 * 3y + y * 3y) = 54 5y + (6y + 3y2) = 54 Combine like terms: 5y + 6y = 11y 11y + 3y2 = 54 Solving 11y + 3y2 = 54 Solving for variable 'y'. Reorder the terms: -54 + 11y + 3y2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + 11y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -18 + 3.666666667y + y2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + 3.666666667y + 18 + y2 = 0 + 18 Reorder the terms: -18 + 18 + 3.666666667y + y2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 3.666666667y + y2 = 0 + 18 3.666666667y + y2 = 0 + 18 Combine like terms: 0 + 18 = 18 3.666666667y + y2 = 18 The y term is 3.666666667y. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667y + 3.361111114 + y2 = 18 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667y + y2 = 18 + 3.361111114 Combine like terms: 18 + 3.361111114 = 21.361111114 3.361111114 + 3.666666667y + y2 = 21.361111114 Factor a perfect square on the left side: (y + 1.833333334)(y + 1.833333334) = 21.361111114 Calculate the square root of the right side: 4.621808208 Break this problem into two subproblems by setting (y + 1.833333334) equal to 4.621808208 and -4.621808208.Subproblem 1
y + 1.833333334 = 4.621808208 Simplifying y + 1.833333334 = 4.621808208 Reorder the terms: 1.833333334 + y = 4.621808208 Solving 1.833333334 + y = 4.621808208 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + y = 4.621808208 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + y = 4.621808208 + -1.833333334 y = 4.621808208 + -1.833333334 Combine like terms: 4.621808208 + -1.833333334 = 2.788474874 y = 2.788474874 Simplifying y = 2.788474874Subproblem 2
y + 1.833333334 = -4.621808208 Simplifying y + 1.833333334 = -4.621808208 Reorder the terms: 1.833333334 + y = -4.621808208 Solving 1.833333334 + y = -4.621808208 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + y = -4.621808208 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + y = -4.621808208 + -1.833333334 y = -4.621808208 + -1.833333334 Combine like terms: -4.621808208 + -1.833333334 = -6.455141542 y = -6.455141542 Simplifying y = -6.455141542Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.788474874, -6.455141542}
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