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Simplifying -9(1 + -1y) * y + 81(1 + -1y) + -1 = 0 Reorder the terms for easier multiplication: -9y(1 + -1y) + 81(1 + -1y) + -1 = 0 (1 * -9y + -1y * -9y) + 81(1 + -1y) + -1 = 0 (-9y + 9y2) + 81(1 + -1y) + -1 = 0 -9y + 9y2 + (1 * 81 + -1y * 81) + -1 = 0 -9y + 9y2 + (81 + -81y) + -1 = 0 Reorder the terms: 81 + -1 + -9y + -81y + 9y2 = 0 Combine like terms: 81 + -1 = 80 80 + -9y + -81y + 9y2 = 0 Combine like terms: -9y + -81y = -90y 80 + -90y + 9y2 = 0 Solving 80 + -90y + 9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 8.888888889 + -10y + y2 = 0 Move the constant term to the right: Add '-8.888888889' to each side of the equation. 8.888888889 + -10y + -8.888888889 + y2 = 0 + -8.888888889 Reorder the terms: 8.888888889 + -8.888888889 + -10y + y2 = 0 + -8.888888889 Combine like terms: 8.888888889 + -8.888888889 = 0.000000000 0.000000000 + -10y + y2 = 0 + -8.888888889 -10y + y2 = 0 + -8.888888889 Combine like terms: 0 + -8.888888889 = -8.888888889 -10y + y2 = -8.888888889 The y term is -10y. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10y + 25 + y2 = -8.888888889 + 25 Reorder the terms: 25 + -10y + y2 = -8.888888889 + 25 Combine like terms: -8.888888889 + 25 = 16.111111111 25 + -10y + y2 = 16.111111111 Factor a perfect square on the left side: (y + -5)(y + -5) = 16.111111111 Calculate the square root of the right side: 4.01386486 Break this problem into two subproblems by setting (y + -5) equal to 4.01386486 and -4.01386486.Subproblem 1
y + -5 = 4.01386486 Simplifying y + -5 = 4.01386486 Reorder the terms: -5 + y = 4.01386486 Solving -5 + y = 4.01386486 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = 4.01386486 + 5 Combine like terms: -5 + 5 = 0 0 + y = 4.01386486 + 5 y = 4.01386486 + 5 Combine like terms: 4.01386486 + 5 = 9.01386486 y = 9.01386486 Simplifying y = 9.01386486Subproblem 2
y + -5 = -4.01386486 Simplifying y + -5 = -4.01386486 Reorder the terms: -5 + y = -4.01386486 Solving -5 + y = -4.01386486 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = -4.01386486 + 5 Combine like terms: -5 + 5 = 0 0 + y = -4.01386486 + 5 y = -4.01386486 + 5 Combine like terms: -4.01386486 + 5 = 0.98613514 y = 0.98613514 Simplifying y = 0.98613514Solution
The solution to the problem is based on the solutions from the subproblems. y = {9.01386486, 0.98613514}
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