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Simplifying y2 + 2y + 5y = -5 Reorder the terms: 2y + 5y + y2 = -5 Combine like terms: 2y + 5y = 7y 7y + y2 = -5 Solving 7y + y2 = -5 Solving for variable 'y'. Reorder the terms: 5 + 7y + y2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 7y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 7y + -5 + y2 = 0 + -5 Reorder the terms: 5 + -5 + 7y + y2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 7y + y2 = 0 + -5 7y + y2 = 0 + -5 Combine like terms: 0 + -5 = -5 7y + y2 = -5 The y term is 7y. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7y + 12.25 + y2 = -5 + 12.25 Reorder the terms: 12.25 + 7y + y2 = -5 + 12.25 Combine like terms: -5 + 12.25 = 7.25 12.25 + 7y + y2 = 7.25 Factor a perfect square on the left side: (y + 3.5)(y + 3.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (y + 3.5) equal to 2.692582404 and -2.692582404.Subproblem 1
y + 3.5 = 2.692582404 Simplifying y + 3.5 = 2.692582404 Reorder the terms: 3.5 + y = 2.692582404 Solving 3.5 + y = 2.692582404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = 2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = 2.692582404 + -3.5 y = 2.692582404 + -3.5 Combine like terms: 2.692582404 + -3.5 = -0.807417596 y = -0.807417596 Simplifying y = -0.807417596Subproblem 2
y + 3.5 = -2.692582404 Simplifying y + 3.5 = -2.692582404 Reorder the terms: 3.5 + y = -2.692582404 Solving 3.5 + y = -2.692582404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = -2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = -2.692582404 + -3.5 y = -2.692582404 + -3.5 Combine like terms: -2.692582404 + -3.5 = -6.192582404 y = -6.192582404 Simplifying y = -6.192582404Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.807417596, -6.192582404}
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