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Simplifying 2y2 + 17y + 34 = y2 + 16 Reorder the terms: 34 + 17y + 2y2 = y2 + 16 Reorder the terms: 34 + 17y + 2y2 = 16 + y2 Solving 34 + 17y + 2y2 = 16 + y2 Solving for variable 'y'. Reorder the terms: 34 + -16 + 17y + 2y2 + -1y2 = 16 + y2 + -16 + -1y2 Combine like terms: 34 + -16 = 18 18 + 17y + 2y2 + -1y2 = 16 + y2 + -16 + -1y2 Combine like terms: 2y2 + -1y2 = 1y2 18 + 17y + 1y2 = 16 + y2 + -16 + -1y2 Reorder the terms: 18 + 17y + 1y2 = 16 + -16 + y2 + -1y2 Combine like terms: 16 + -16 = 0 18 + 17y + 1y2 = 0 + y2 + -1y2 18 + 17y + 1y2 = y2 + -1y2 Combine like terms: y2 + -1y2 = 0 18 + 17y + 1y2 = 0 Begin completing the square. Move the constant term to the right: Add '-18' to each side of the equation. 18 + 17y + -18 + y2 = 0 + -18 Reorder the terms: 18 + -18 + 17y + y2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + 17y + y2 = 0 + -18 17y + y2 = 0 + -18 Combine like terms: 0 + -18 = -18 17y + y2 = -18 The y term is 17y. Take half its coefficient (8.5). Square it (72.25) and add it to both sides. Add '72.25' to each side of the equation. 17y + 72.25 + y2 = -18 + 72.25 Reorder the terms: 72.25 + 17y + y2 = -18 + 72.25 Combine like terms: -18 + 72.25 = 54.25 72.25 + 17y + y2 = 54.25 Factor a perfect square on the left side: (y + 8.5)(y + 8.5) = 54.25 Calculate the square root of the right side: 7.365459931 Break this problem into two subproblems by setting (y + 8.5) equal to 7.365459931 and -7.365459931.Subproblem 1
y + 8.5 = 7.365459931 Simplifying y + 8.5 = 7.365459931 Reorder the terms: 8.5 + y = 7.365459931 Solving 8.5 + y = 7.365459931 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + y = 7.365459931 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + y = 7.365459931 + -8.5 y = 7.365459931 + -8.5 Combine like terms: 7.365459931 + -8.5 = -1.134540069 y = -1.134540069 Simplifying y = -1.134540069Subproblem 2
y + 8.5 = -7.365459931 Simplifying y + 8.5 = -7.365459931 Reorder the terms: 8.5 + y = -7.365459931 Solving 8.5 + y = -7.365459931 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + y = -7.365459931 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + y = -7.365459931 + -8.5 y = -7.365459931 + -8.5 Combine like terms: -7.365459931 + -8.5 = -15.865459931 y = -15.865459931 Simplifying y = -15.865459931Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.134540069, -15.865459931}
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