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Simplifying 3y2 + y = 1 Reorder the terms: y + 3y2 = 1 Solving y + 3y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + y + 3y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + 0.3333333333y + y2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.3333333333y + 0.3333333333 + y2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.3333333333y + y2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.3333333333y + y2 = 0 + 0.3333333333 0.3333333333y + y2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.3333333333y + y2 = 0.3333333333 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. 0.3333333333y + 0.25 + y2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + 0.3333333333y + y2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + 0.3333333333y + y2 = 0.5833333333 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (y + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
y + 0.5 = 0.763762616 Simplifying y + 0.5 = 0.763762616 Reorder the terms: 0.5 + y = 0.763762616 Solving 0.5 + y = 0.763762616 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 = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 0.763762616 + -0.5 y = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 y = 0.263762616 Simplifying y = 0.263762616Subproblem 2
y + 0.5 = -0.763762616 Simplifying y + 0.5 = -0.763762616 Reorder the terms: 0.5 + y = -0.763762616 Solving 0.5 + y = -0.763762616 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 = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -0.763762616 + -0.5 y = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 y = -1.263762616 Simplifying y = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.263762616, -1.263762616}
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