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Simplifying y2 + (y + 3)(y + -5) = 0 Reorder the terms: y2 + (3 + y)(y + -5) = 0 Reorder the terms: y2 + (3 + y)(-5 + y) = 0 Multiply (3 + y) * (-5 + y) y2 + (3(-5 + y) + y(-5 + y)) = 0 y2 + ((-5 * 3 + y * 3) + y(-5 + y)) = 0 y2 + ((-15 + 3y) + y(-5 + y)) = 0 y2 + (-15 + 3y + (-5 * y + y * y)) = 0 y2 + (-15 + 3y + (-5y + y2)) = 0 Combine like terms: 3y + -5y = -2y y2 + (-15 + -2y + y2) = 0 Reorder the terms: -15 + -2y + y2 + y2 = 0 Combine like terms: y2 + y2 = 2y2 -15 + -2y + 2y2 = 0 Solving -15 + -2y + 2y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + -1y + y2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -1y + 7.5 + y2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -1y + y2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -1y + y2 = 0 + 7.5 -1y + y2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -1y + y2 = 7.5 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 = 7.5 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 7.5 + 0.25 Combine like terms: 7.5 + 0.25 = 7.75 0.25 + -1y + y2 = 7.75 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (y + -0.5) equal to 2.783882181 and -2.783882181.Subproblem 1
y + -0.5 = 2.783882181 Simplifying y + -0.5 = 2.783882181 Reorder the terms: -0.5 + y = 2.783882181 Solving -0.5 + y = 2.783882181 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 = 2.783882181 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 2.783882181 + 0.5 y = 2.783882181 + 0.5 Combine like terms: 2.783882181 + 0.5 = 3.283882181 y = 3.283882181 Simplifying y = 3.283882181Subproblem 2
y + -0.5 = -2.783882181 Simplifying y + -0.5 = -2.783882181 Reorder the terms: -0.5 + y = -2.783882181 Solving -0.5 + y = -2.783882181 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 = -2.783882181 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -2.783882181 + 0.5 y = -2.783882181 + 0.5 Combine like terms: -2.783882181 + 0.5 = -2.283882181 y = -2.283882181 Simplifying y = -2.283882181Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.283882181, -2.283882181}
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