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Simplifying (9 + -1y)(9y) = 1 Remove parenthesis around (9y) (9 + -1y) * 9y = 1 Reorder the terms for easier multiplication: 9y(9 + -1y) = 1 (9 * 9y + -1y * 9y) = 1 (81y + -9y2) = 1 Solving 81y + -9y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + 81y + -9y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 81y + -9y2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. 0.1111111111 + -9y + y2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + -9y + -0.1111111111 + y2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + -9y + y2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + -9y + y2 = 0 + -0.1111111111 -9y + y2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 -9y + y2 = -0.1111111111 The y term is -9y. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9y + 20.25 + y2 = -0.1111111111 + 20.25 Reorder the terms: 20.25 + -9y + y2 = -0.1111111111 + 20.25 Combine like terms: -0.1111111111 + 20.25 = 20.1388888889 20.25 + -9y + y2 = 20.1388888889 Factor a perfect square on the left side: (y + -4.5)(y + -4.5) = 20.1388888889 Calculate the square root of the right side: 4.487637339 Break this problem into two subproblems by setting (y + -4.5) equal to 4.487637339 and -4.487637339.Subproblem 1
y + -4.5 = 4.487637339 Simplifying y + -4.5 = 4.487637339 Reorder the terms: -4.5 + y = 4.487637339 Solving -4.5 + y = 4.487637339 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = 4.487637339 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = 4.487637339 + 4.5 y = 4.487637339 + 4.5 Combine like terms: 4.487637339 + 4.5 = 8.987637339 y = 8.987637339 Simplifying y = 8.987637339Subproblem 2
y + -4.5 = -4.487637339 Simplifying y + -4.5 = -4.487637339 Reorder the terms: -4.5 + y = -4.487637339 Solving -4.5 + y = -4.487637339 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = -4.487637339 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = -4.487637339 + 4.5 y = -4.487637339 + 4.5 Combine like terms: -4.487637339 + 4.5 = 0.012362661 y = 0.012362661 Simplifying y = 0.012362661Solution
The solution to the problem is based on the solutions from the subproblems. y = {8.987637339, 0.012362661}
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