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Simplifying y + -19y2 = -60 Solving y + -19y2 = -60 Solving for variable 'y'. Reorder the terms: 60 + y + -19y2 = -60 + 60 Combine like terms: -60 + 60 = 0 60 + y + -19y2 = 0 Begin completing the square. Divide all terms by -19 the coefficient of the squared term: Divide each side by '-19'. -3.157894737 + -0.05263157895y + y2 = 0 Move the constant term to the right: Add '3.157894737' to each side of the equation. -3.157894737 + -0.05263157895y + 3.157894737 + y2 = 0 + 3.157894737 Reorder the terms: -3.157894737 + 3.157894737 + -0.05263157895y + y2 = 0 + 3.157894737 Combine like terms: -3.157894737 + 3.157894737 = 0.000000000 0.000000000 + -0.05263157895y + y2 = 0 + 3.157894737 -0.05263157895y + y2 = 0 + 3.157894737 Combine like terms: 0 + 3.157894737 = 3.157894737 -0.05263157895y + y2 = 3.157894737 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. -0.05263157895y + 0.25 + y2 = 3.157894737 + 0.25 Reorder the terms: 0.25 + -0.05263157895y + y2 = 3.157894737 + 0.25 Combine like terms: 3.157894737 + 0.25 = 3.407894737 0.25 + -0.05263157895y + y2 = 3.407894737 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 3.407894737 Calculate the square root of the right side: 1.846048411 Break this problem into two subproblems by setting (y + 0.5) equal to 1.846048411 and -1.846048411.Subproblem 1
y + 0.5 = 1.846048411 Simplifying y + 0.5 = 1.846048411 Reorder the terms: 0.5 + y = 1.846048411 Solving 0.5 + y = 1.846048411 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 = 1.846048411 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.846048411 + -0.5 y = 1.846048411 + -0.5 Combine like terms: 1.846048411 + -0.5 = 1.346048411 y = 1.346048411 Simplifying y = 1.346048411Subproblem 2
y + 0.5 = -1.846048411 Simplifying y + 0.5 = -1.846048411 Reorder the terms: 0.5 + y = -1.846048411 Solving 0.5 + y = -1.846048411 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 = -1.846048411 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.846048411 + -0.5 y = -1.846048411 + -0.5 Combine like terms: -1.846048411 + -0.5 = -2.346048411 y = -2.346048411 Simplifying y = -2.346048411Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.346048411, -2.346048411}
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