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Simplifying 12 + y = 34 + -1y2 Solving 12 + y = 34 + -1y2 Solving for variable 'y'. Reorder the terms: 12 + -34 + y + y2 = 34 + -1y2 + -34 + y2 Combine like terms: 12 + -34 = -22 -22 + y + y2 = 34 + -1y2 + -34 + y2 Reorder the terms: -22 + y + y2 = 34 + -34 + -1y2 + y2 Combine like terms: 34 + -34 = 0 -22 + y + y2 = 0 + -1y2 + y2 -22 + y + y2 = -1y2 + y2 Combine like terms: -1y2 + y2 = 0 -22 + y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '22' to each side of the equation. -22 + y + 22 + y2 = 0 + 22 Reorder the terms: -22 + 22 + y + y2 = 0 + 22 Combine like terms: -22 + 22 = 0 0 + y + y2 = 0 + 22 y + y2 = 0 + 22 Combine like terms: 0 + 22 = 22 y + y2 = 22 The y term is y. 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. y + 0.25 + y2 = 22 + 0.25 Reorder the terms: 0.25 + y + y2 = 22 + 0.25 Combine like terms: 22 + 0.25 = 22.25 0.25 + y + y2 = 22.25 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (y + 0.5) equal to 4.716990566 and -4.716990566.Subproblem 1
y + 0.5 = 4.716990566 Simplifying y + 0.5 = 4.716990566 Reorder the terms: 0.5 + y = 4.716990566 Solving 0.5 + y = 4.716990566 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 = 4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 4.716990566 + -0.5 y = 4.716990566 + -0.5 Combine like terms: 4.716990566 + -0.5 = 4.216990566 y = 4.216990566 Simplifying y = 4.216990566Subproblem 2
y + 0.5 = -4.716990566 Simplifying y + 0.5 = -4.716990566 Reorder the terms: 0.5 + y = -4.716990566 Solving 0.5 + y = -4.716990566 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 = -4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -4.716990566 + -0.5 y = -4.716990566 + -0.5 Combine like terms: -4.716990566 + -0.5 = -5.216990566 y = -5.216990566 Simplifying y = -5.216990566Solution
The solution to the problem is based on the solutions from the subproblems. y = {4.216990566, -5.216990566}
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