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Simplifying 3y(y + -2) = 11 Reorder the terms: 3y(-2 + y) = 11 (-2 * 3y + y * 3y) = 11 (-6y + 3y2) = 11 Solving -6y + 3y2 = 11 Solving for variable 'y'. Reorder the terms: -11 + -6y + 3y2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + -6y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.666666667 + -2y + y2 = 0 Move the constant term to the right: Add '3.666666667' to each side of the equation. -3.666666667 + -2y + 3.666666667 + y2 = 0 + 3.666666667 Reorder the terms: -3.666666667 + 3.666666667 + -2y + y2 = 0 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + -2y + y2 = 0 + 3.666666667 -2y + y2 = 0 + 3.666666667 Combine like terms: 0 + 3.666666667 = 3.666666667 -2y + y2 = 3.666666667 The y term is -2y. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2y + 1 + y2 = 3.666666667 + 1 Reorder the terms: 1 + -2y + y2 = 3.666666667 + 1 Combine like terms: 3.666666667 + 1 = 4.666666667 1 + -2y + y2 = 4.666666667 Factor a perfect square on the left side: (y + -1)(y + -1) = 4.666666667 Calculate the square root of the right side: 2.1602469 Break this problem into two subproblems by setting (y + -1) equal to 2.1602469 and -2.1602469.Subproblem 1
y + -1 = 2.1602469 Simplifying y + -1 = 2.1602469 Reorder the terms: -1 + y = 2.1602469 Solving -1 + y = 2.1602469 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + y = 2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + y = 2.1602469 + 1 y = 2.1602469 + 1 Combine like terms: 2.1602469 + 1 = 3.1602469 y = 3.1602469 Simplifying y = 3.1602469Subproblem 2
y + -1 = -2.1602469 Simplifying y + -1 = -2.1602469 Reorder the terms: -1 + y = -2.1602469 Solving -1 + y = -2.1602469 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + y = -2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + y = -2.1602469 + 1 y = -2.1602469 + 1 Combine like terms: -2.1602469 + 1 = -1.1602469 y = -1.1602469 Simplifying y = -1.1602469Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.1602469, -1.1602469}
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