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Simplifying (y + 8)(y + -9) = 52 Reorder the terms: (8 + y)(y + -9) = 52 Reorder the terms: (8 + y)(-9 + y) = 52 Multiply (8 + y) * (-9 + y) (8(-9 + y) + y(-9 + y)) = 52 ((-9 * 8 + y * 8) + y(-9 + y)) = 52 ((-72 + 8y) + y(-9 + y)) = 52 (-72 + 8y + (-9 * y + y * y)) = 52 (-72 + 8y + (-9y + y2)) = 52 Combine like terms: 8y + -9y = -1y (-72 + -1y + y2) = 52 Solving -72 + -1y + y2 = 52 Solving for variable 'y'. Reorder the terms: -72 + -52 + -1y + y2 = 52 + -52 Combine like terms: -72 + -52 = -124 -124 + -1y + y2 = 52 + -52 Combine like terms: 52 + -52 = 0 -124 + -1y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '124' to each side of the equation. -124 + -1y + 124 + y2 = 0 + 124 Reorder the terms: -124 + 124 + -1y + y2 = 0 + 124 Combine like terms: -124 + 124 = 0 0 + -1y + y2 = 0 + 124 -1y + y2 = 0 + 124 Combine like terms: 0 + 124 = 124 -1y + y2 = 124 The y term is -1y. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1y + 0.25 + y2 = 124 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 124 + 0.25 Combine like terms: 124 + 0.25 = 124.25 0.25 + -1y + y2 = 124.25 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 124.25 Calculate the square root of the right side: 11.146748405 Break this problem into two subproblems by setting (y + -0.5) equal to 11.146748405 and -11.146748405.Subproblem 1
y + -0.5 = 11.146748405 Simplifying y + -0.5 = 11.146748405 Reorder the terms: -0.5 + y = 11.146748405 Solving -0.5 + y = 11.146748405 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = 11.146748405 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 11.146748405 + 0.5 y = 11.146748405 + 0.5 Combine like terms: 11.146748405 + 0.5 = 11.646748405 y = 11.646748405 Simplifying y = 11.646748405Subproblem 2
y + -0.5 = -11.146748405 Simplifying y + -0.5 = -11.146748405 Reorder the terms: -0.5 + y = -11.146748405 Solving -0.5 + y = -11.146748405 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = -11.146748405 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -11.146748405 + 0.5 y = -11.146748405 + 0.5 Combine like terms: -11.146748405 + 0.5 = -10.646748405 y = -10.646748405 Simplifying y = -10.646748405Solution
The solution to the problem is based on the solutions from the subproblems. y = {11.646748405, -10.646748405}
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