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Simplifying y2 + 11y + 45 = 23 Reorder the terms: 45 + 11y + y2 = 23 Solving 45 + 11y + y2 = 23 Solving for variable 'y'. Reorder the terms: 45 + -23 + 11y + y2 = 23 + -23 Combine like terms: 45 + -23 = 22 22 + 11y + y2 = 23 + -23 Combine like terms: 23 + -23 = 0 22 + 11y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-22' to each side of the equation. 22 + 11y + -22 + y2 = 0 + -22 Reorder the terms: 22 + -22 + 11y + y2 = 0 + -22 Combine like terms: 22 + -22 = 0 0 + 11y + y2 = 0 + -22 11y + y2 = 0 + -22 Combine like terms: 0 + -22 = -22 11y + y2 = -22 The y term is 11y. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11y + 30.25 + y2 = -22 + 30.25 Reorder the terms: 30.25 + 11y + y2 = -22 + 30.25 Combine like terms: -22 + 30.25 = 8.25 30.25 + 11y + y2 = 8.25 Factor a perfect square on the left side: (y + 5.5)(y + 5.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (y + 5.5) equal to 2.872281323 and -2.872281323.Subproblem 1
y + 5.5 = 2.872281323 Simplifying y + 5.5 = 2.872281323 Reorder the terms: 5.5 + y = 2.872281323 Solving 5.5 + y = 2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + y = 2.872281323 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + y = 2.872281323 + -5.5 y = 2.872281323 + -5.5 Combine like terms: 2.872281323 + -5.5 = -2.627718677 y = -2.627718677 Simplifying y = -2.627718677Subproblem 2
y + 5.5 = -2.872281323 Simplifying y + 5.5 = -2.872281323 Reorder the terms: 5.5 + y = -2.872281323 Solving 5.5 + y = -2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + y = -2.872281323 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + y = -2.872281323 + -5.5 y = -2.872281323 + -5.5 Combine like terms: -2.872281323 + -5.5 = -8.372281323 y = -8.372281323 Simplifying y = -8.372281323Solution
The solution to the problem is based on the solutions from the subproblems. y = {-2.627718677, -8.372281323}
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