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Simplifying (3y + 12)(-2y + 2) = 13 Reorder the terms: (12 + 3y)(-2y + 2) = 13 Reorder the terms: (12 + 3y)(2 + -2y) = 13 Multiply (12 + 3y) * (2 + -2y) (12(2 + -2y) + 3y * (2 + -2y)) = 13 ((2 * 12 + -2y * 12) + 3y * (2 + -2y)) = 13 ((24 + -24y) + 3y * (2 + -2y)) = 13 (24 + -24y + (2 * 3y + -2y * 3y)) = 13 (24 + -24y + (6y + -6y2)) = 13 Combine like terms: -24y + 6y = -18y (24 + -18y + -6y2) = 13 Solving 24 + -18y + -6y2 = 13 Solving for variable 'y'. Reorder the terms: 24 + -13 + -18y + -6y2 = 13 + -13 Combine like terms: 24 + -13 = 11 11 + -18y + -6y2 = 13 + -13 Combine like terms: 13 + -13 = 0 11 + -18y + -6y2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.833333333 + 3y + y2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + 3y + 1.833333333 + y2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + 3y + y2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + 3y + y2 = 0 + 1.833333333 3y + y2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 3y + y2 = 1.833333333 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = 1.833333333 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 1.833333333 + 2.25 Combine like terms: 1.833333333 + 2.25 = 4.083333333 2.25 + 3y + y2 = 4.083333333 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 4.083333333 Calculate the square root of the right side: 2.020725942 Break this problem into two subproblems by setting (y + 1.5) equal to 2.020725942 and -2.020725942.Subproblem 1
y + 1.5 = 2.020725942 Simplifying y + 1.5 = 2.020725942 Reorder the terms: 1.5 + y = 2.020725942 Solving 1.5 + y = 2.020725942 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 2.020725942 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 2.020725942 + -1.5 y = 2.020725942 + -1.5 Combine like terms: 2.020725942 + -1.5 = 0.520725942 y = 0.520725942 Simplifying y = 0.520725942Subproblem 2
y + 1.5 = -2.020725942 Simplifying y + 1.5 = -2.020725942 Reorder the terms: 1.5 + y = -2.020725942 Solving 1.5 + y = -2.020725942 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -2.020725942 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -2.020725942 + -1.5 y = -2.020725942 + -1.5 Combine like terms: -2.020725942 + -1.5 = -3.520725942 y = -3.520725942 Simplifying y = -3.520725942Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.520725942, -3.520725942}
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