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Simplifying 10y2 + 41y = -9 Reorder the terms: 41y + 10y2 = -9 Solving 41y + 10y2 = -9 Solving for variable 'y'. Reorder the terms: 9 + 41y + 10y2 = -9 + 9 Combine like terms: -9 + 9 = 0 9 + 41y + 10y2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.9 + 4.1y + y2 = 0 Move the constant term to the right: Add '-0.9' to each side of the equation. 0.9 + 4.1y + -0.9 + y2 = 0 + -0.9 Reorder the terms: 0.9 + -0.9 + 4.1y + y2 = 0 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + 4.1y + y2 = 0 + -0.9 4.1y + y2 = 0 + -0.9 Combine like terms: 0 + -0.9 = -0.9 4.1y + y2 = -0.9 The y term is 4.1y. Take half its coefficient (2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. 4.1y + 4.2025 + y2 = -0.9 + 4.2025 Reorder the terms: 4.2025 + 4.1y + y2 = -0.9 + 4.2025 Combine like terms: -0.9 + 4.2025 = 3.3025 4.2025 + 4.1y + y2 = 3.3025 Factor a perfect square on the left side: (y + 2.05)(y + 2.05) = 3.3025 Calculate the square root of the right side: 1.817278185 Break this problem into two subproblems by setting (y + 2.05) equal to 1.817278185 and -1.817278185.Subproblem 1
y + 2.05 = 1.817278185 Simplifying y + 2.05 = 1.817278185 Reorder the terms: 2.05 + y = 1.817278185 Solving 2.05 + y = 1.817278185 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + y = 1.817278185 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + y = 1.817278185 + -2.05 y = 1.817278185 + -2.05 Combine like terms: 1.817278185 + -2.05 = -0.232721815 y = -0.232721815 Simplifying y = -0.232721815Subproblem 2
y + 2.05 = -1.817278185 Simplifying y + 2.05 = -1.817278185 Reorder the terms: 2.05 + y = -1.817278185 Solving 2.05 + y = -1.817278185 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + y = -1.817278185 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + y = -1.817278185 + -2.05 y = -1.817278185 + -2.05 Combine like terms: -1.817278185 + -2.05 = -3.867278185 y = -3.867278185 Simplifying y = -3.867278185Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.232721815, -3.867278185}
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