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Simplifying (y + 5)(y + -7) = 20 Reorder the terms: (5 + y)(y + -7) = 20 Reorder the terms: (5 + y)(-7 + y) = 20 Multiply (5 + y) * (-7 + y) (5(-7 + y) + y(-7 + y)) = 20 ((-7 * 5 + y * 5) + y(-7 + y)) = 20 ((-35 + 5y) + y(-7 + y)) = 20 (-35 + 5y + (-7 * y + y * y)) = 20 (-35 + 5y + (-7y + y2)) = 20 Combine like terms: 5y + -7y = -2y (-35 + -2y + y2) = 20 Solving -35 + -2y + y2 = 20 Solving for variable 'y'. Reorder the terms: -35 + -20 + -2y + y2 = 20 + -20 Combine like terms: -35 + -20 = -55 -55 + -2y + y2 = 20 + -20 Combine like terms: 20 + -20 = 0 -55 + -2y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '55' to each side of the equation. -55 + -2y + 55 + y2 = 0 + 55 Reorder the terms: -55 + 55 + -2y + y2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + -2y + y2 = 0 + 55 -2y + y2 = 0 + 55 Combine like terms: 0 + 55 = 55 -2y + y2 = 55 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 = 55 + 1 Reorder the terms: 1 + -2y + y2 = 55 + 1 Combine like terms: 55 + 1 = 56 1 + -2y + y2 = 56 Factor a perfect square on the left side: (y + -1)(y + -1) = 56 Calculate the square root of the right side: 7.483314774 Break this problem into two subproblems by setting (y + -1) equal to 7.483314774 and -7.483314774.Subproblem 1
y + -1 = 7.483314774 Simplifying y + -1 = 7.483314774 Reorder the terms: -1 + y = 7.483314774 Solving -1 + y = 7.483314774 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 = 7.483314774 + 1 Combine like terms: -1 + 1 = 0 0 + y = 7.483314774 + 1 y = 7.483314774 + 1 Combine like terms: 7.483314774 + 1 = 8.483314774 y = 8.483314774 Simplifying y = 8.483314774Subproblem 2
y + -1 = -7.483314774 Simplifying y + -1 = -7.483314774 Reorder the terms: -1 + y = -7.483314774 Solving -1 + y = -7.483314774 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 = -7.483314774 + 1 Combine like terms: -1 + 1 = 0 0 + y = -7.483314774 + 1 y = -7.483314774 + 1 Combine like terms: -7.483314774 + 1 = -6.483314774 y = -6.483314774 Simplifying y = -6.483314774Solution
The solution to the problem is based on the solutions from the subproblems. y = {8.483314774, -6.483314774}
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