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Simplifying (3y + 11)(6y + -1) = 180 Reorder the terms: (11 + 3y)(6y + -1) = 180 Reorder the terms: (11 + 3y)(-1 + 6y) = 180 Multiply (11 + 3y) * (-1 + 6y) (11(-1 + 6y) + 3y * (-1 + 6y)) = 180 ((-1 * 11 + 6y * 11) + 3y * (-1 + 6y)) = 180 ((-11 + 66y) + 3y * (-1 + 6y)) = 180 (-11 + 66y + (-1 * 3y + 6y * 3y)) = 180 (-11 + 66y + (-3y + 18y2)) = 180 Combine like terms: 66y + -3y = 63y (-11 + 63y + 18y2) = 180 Solving -11 + 63y + 18y2 = 180 Solving for variable 'y'. Reorder the terms: -11 + -180 + 63y + 18y2 = 180 + -180 Combine like terms: -11 + -180 = -191 -191 + 63y + 18y2 = 180 + -180 Combine like terms: 180 + -180 = 0 -191 + 63y + 18y2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -10.61111111 + 3.5y + y2 = 0 Move the constant term to the right: Add '10.61111111' to each side of the equation. -10.61111111 + 3.5y + 10.61111111 + y2 = 0 + 10.61111111 Reorder the terms: -10.61111111 + 10.61111111 + 3.5y + y2 = 0 + 10.61111111 Combine like terms: -10.61111111 + 10.61111111 = 0.00000000 0.00000000 + 3.5y + y2 = 0 + 10.61111111 3.5y + y2 = 0 + 10.61111111 Combine like terms: 0 + 10.61111111 = 10.61111111 3.5y + y2 = 10.61111111 The y term is 3.5y. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5y + 3.0625 + y2 = 10.61111111 + 3.0625 Reorder the terms: 3.0625 + 3.5y + y2 = 10.61111111 + 3.0625 Combine like terms: 10.61111111 + 3.0625 = 13.67361111 3.0625 + 3.5y + y2 = 13.67361111 Factor a perfect square on the left side: (y + 1.75)(y + 1.75) = 13.67361111 Calculate the square root of the right side: 3.697784622 Break this problem into two subproblems by setting (y + 1.75) equal to 3.697784622 and -3.697784622.Subproblem 1
y + 1.75 = 3.697784622 Simplifying y + 1.75 = 3.697784622 Reorder the terms: 1.75 + y = 3.697784622 Solving 1.75 + y = 3.697784622 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = 3.697784622 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 3.697784622 + -1.75 y = 3.697784622 + -1.75 Combine like terms: 3.697784622 + -1.75 = 1.947784622 y = 1.947784622 Simplifying y = 1.947784622Subproblem 2
y + 1.75 = -3.697784622 Simplifying y + 1.75 = -3.697784622 Reorder the terms: 1.75 + y = -3.697784622 Solving 1.75 + y = -3.697784622 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = -3.697784622 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -3.697784622 + -1.75 y = -3.697784622 + -1.75 Combine like terms: -3.697784622 + -1.75 = -5.447784622 y = -5.447784622 Simplifying y = -5.447784622Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.947784622, -5.447784622}
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