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Simplifying 9(1 + -1y)(9 + -1y) = 1 Multiply (1 + -1y) * (9 + -1y) 9(1(9 + -1y) + -1y * (9 + -1y)) = 1 9((9 * 1 + -1y * 1) + -1y * (9 + -1y)) = 1 9((9 + -1y) + -1y * (9 + -1y)) = 1 9(9 + -1y + (9 * -1y + -1y * -1y)) = 1 9(9 + -1y + (-9y + 1y2)) = 1 Combine like terms: -1y + -9y = -10y 9(9 + -10y + 1y2) = 1 (9 * 9 + -10y * 9 + 1y2 * 9) = 1 (81 + -90y + 9y2) = 1 Solving 81 + -90y + 9y2 = 1 Solving for variable 'y'. Reorder the terms: 81 + -1 + -90y + 9y2 = 1 + -1 Combine like terms: 81 + -1 = 80 80 + -90y + 9y2 = 1 + -1 Combine like terms: 1 + -1 = 0 80 + -90y + 9y2 = 0 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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