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Simplifying -9(1 + y) * y + 81(1 + y) + -1 = 0 Reorder the terms for easier multiplication: -9y(1 + y) + 81(1 + y) + -1 = 0 (1 * -9y + y * -9y) + 81(1 + y) + -1 = 0 (-9y + -9y2) + 81(1 + y) + -1 = 0 -9y + -9y2 + (1 * 81 + y * 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 = 72y 80 + 72y + -9y2 = 0 Solving 80 + 72y + -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 + -8y + y2 = 0 Move the constant term to the right: Add '8.888888889' to each side of the equation. -8.888888889 + -8y + 8.888888889 + y2 = 0 + 8.888888889 Reorder the terms: -8.888888889 + 8.888888889 + -8y + y2 = 0 + 8.888888889 Combine like terms: -8.888888889 + 8.888888889 = 0.000000000 0.000000000 + -8y + y2 = 0 + 8.888888889 -8y + y2 = 0 + 8.888888889 Combine like terms: 0 + 8.888888889 = 8.888888889 -8y + y2 = 8.888888889 The y term is -8y. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8y + 16 + y2 = 8.888888889 + 16 Reorder the terms: 16 + -8y + y2 = 8.888888889 + 16 Combine like terms: 8.888888889 + 16 = 24.888888889 16 + -8y + y2 = 24.888888889 Factor a perfect square on the left side: (y + -4)(y + -4) = 24.888888889 Calculate the square root of the right side: 4.988876516 Break this problem into two subproblems by setting (y + -4) equal to 4.988876516 and -4.988876516.Subproblem 1
y + -4 = 4.988876516 Simplifying y + -4 = 4.988876516 Reorder the terms: -4 + y = 4.988876516 Solving -4 + y = 4.988876516 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + y = 4.988876516 + 4 Combine like terms: -4 + 4 = 0 0 + y = 4.988876516 + 4 y = 4.988876516 + 4 Combine like terms: 4.988876516 + 4 = 8.988876516 y = 8.988876516 Simplifying y = 8.988876516Subproblem 2
y + -4 = -4.988876516 Simplifying y + -4 = -4.988876516 Reorder the terms: -4 + y = -4.988876516 Solving -4 + y = -4.988876516 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + y = -4.988876516 + 4 Combine like terms: -4 + 4 = 0 0 + y = -4.988876516 + 4 y = -4.988876516 + 4 Combine like terms: -4.988876516 + 4 = -0.988876516 y = -0.988876516 Simplifying y = -0.988876516Solution
The solution to the problem is based on the solutions from the subproblems. y = {8.988876516, -0.988876516}
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