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Simplifying y2 + 4 = 2y2 + 13y + 18 Reorder the terms: 4 + y2 = 2y2 + 13y + 18 Reorder the terms: 4 + y2 = 18 + 13y + 2y2 Solving 4 + y2 = 18 + 13y + 2y2 Solving for variable 'y'. Reorder the terms: 4 + -18 + -13y + y2 + -2y2 = 18 + 13y + 2y2 + -18 + -13y + -2y2 Combine like terms: 4 + -18 = -14 -14 + -13y + y2 + -2y2 = 18 + 13y + 2y2 + -18 + -13y + -2y2 Combine like terms: y2 + -2y2 = -1y2 -14 + -13y + -1y2 = 18 + 13y + 2y2 + -18 + -13y + -2y2 Reorder the terms: -14 + -13y + -1y2 = 18 + -18 + 13y + -13y + 2y2 + -2y2 Combine like terms: 18 + -18 = 0 -14 + -13y + -1y2 = 0 + 13y + -13y + 2y2 + -2y2 -14 + -13y + -1y2 = 13y + -13y + 2y2 + -2y2 Combine like terms: 13y + -13y = 0 -14 + -13y + -1y2 = 0 + 2y2 + -2y2 -14 + -13y + -1y2 = 2y2 + -2y2 Combine like terms: 2y2 + -2y2 = 0 -14 + -13y + -1y2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(14 + 13y + y2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(14 + 13y + y2)' equal to zero and attempt to solve: Simplifying 14 + 13y + y2 = 0 Solving 14 + 13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + 13y + -14 + y2 = 0 + -14 Reorder the terms: 14 + -14 + 13y + y2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + 13y + y2 = 0 + -14 13y + y2 = 0 + -14 Combine like terms: 0 + -14 = -14 13y + y2 = -14 The y term is 13y. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13y + 42.25 + y2 = -14 + 42.25 Reorder the terms: 42.25 + 13y + y2 = -14 + 42.25 Combine like terms: -14 + 42.25 = 28.25 42.25 + 13y + y2 = 28.25 Factor a perfect square on the left side: (y + 6.5)(y + 6.5) = 28.25 Calculate the square root of the right side: 5.315072906 Break this problem into two subproblems by setting (y + 6.5) equal to 5.315072906 and -5.315072906.Subproblem 1
y + 6.5 = 5.315072906 Simplifying y + 6.5 = 5.315072906 Reorder the terms: 6.5 + y = 5.315072906 Solving 6.5 + y = 5.315072906 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + y = 5.315072906 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + y = 5.315072906 + -6.5 y = 5.315072906 + -6.5 Combine like terms: 5.315072906 + -6.5 = -1.184927094 y = -1.184927094 Simplifying y = -1.184927094Subproblem 2
y + 6.5 = -5.315072906 Simplifying y + 6.5 = -5.315072906 Reorder the terms: 6.5 + y = -5.315072906 Solving 6.5 + y = -5.315072906 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + y = -5.315072906 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + y = -5.315072906 + -6.5 y = -5.315072906 + -6.5 Combine like terms: -5.315072906 + -6.5 = -11.815072906 y = -11.815072906 Simplifying y = -11.815072906Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.184927094, -11.815072906}Solution
y = {-1.184927094, -11.815072906}
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