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Simplifying (y + 3)(2y + -2) = 1 Reorder the terms: (3 + y)(2y + -2) = 1 Reorder the terms: (3 + y)(-2 + 2y) = 1 Multiply (3 + y) * (-2 + 2y) (3(-2 + 2y) + y(-2 + 2y)) = 1 ((-2 * 3 + 2y * 3) + y(-2 + 2y)) = 1 ((-6 + 6y) + y(-2 + 2y)) = 1 (-6 + 6y + (-2 * y + 2y * y)) = 1 (-6 + 6y + (-2y + 2y2)) = 1 Combine like terms: 6y + -2y = 4y (-6 + 4y + 2y2) = 1 Solving -6 + 4y + 2y2 = 1 Solving for variable 'y'. Reorder the terms: -6 + -1 + 4y + 2y2 = 1 + -1 Combine like terms: -6 + -1 = -7 -7 + 4y + 2y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -7 + 4y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 2y + y2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 2y + 3.5 + y2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 2y + y2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 2y + y2 = 0 + 3.5 2y + y2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 2y + y2 = 3.5 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 = 3.5 + 1 Reorder the terms: 1 + 2y + y2 = 3.5 + 1 Combine like terms: 3.5 + 1 = 4.5 1 + 2y + y2 = 4.5 Factor a perfect square on the left side: (y + 1)(y + 1) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (y + 1) equal to 2.121320344 and -2.121320344.Subproblem 1
y + 1 = 2.121320344 Simplifying y + 1 = 2.121320344 Reorder the terms: 1 + y = 2.121320344 Solving 1 + y = 2.121320344 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 = 2.121320344 + -1 Combine like terms: 1 + -1 = 0 0 + y = 2.121320344 + -1 y = 2.121320344 + -1 Combine like terms: 2.121320344 + -1 = 1.121320344 y = 1.121320344 Simplifying y = 1.121320344Subproblem 2
y + 1 = -2.121320344 Simplifying y + 1 = -2.121320344 Reorder the terms: 1 + y = -2.121320344 Solving 1 + y = -2.121320344 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 = -2.121320344 + -1 Combine like terms: 1 + -1 = 0 0 + y = -2.121320344 + -1 y = -2.121320344 + -1 Combine like terms: -2.121320344 + -1 = -3.121320344 y = -3.121320344 Simplifying y = -3.121320344Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.121320344, -3.121320344}
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