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Simplifying 9y2 + -11y + 3 = 0 Reorder the terms: 3 + -11y + 9y2 = 0 Solving 3 + -11y + 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'. 0.3333333333 + -1.222222222y + y2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -1.222222222y + -0.3333333333 + y2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -1.222222222y + y2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -1.222222222y + y2 = 0 + -0.3333333333 -1.222222222y + y2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -1.222222222y + y2 = -0.3333333333 The y term is -1.222222222y. Take half its coefficient (-0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. -1.222222222y + 0.3734567900 + y2 = -0.3333333333 + 0.3734567900 Reorder the terms: 0.3734567900 + -1.222222222y + y2 = -0.3333333333 + 0.3734567900 Combine like terms: -0.3333333333 + 0.3734567900 = 0.0401234567 0.3734567900 + -1.222222222y + y2 = 0.0401234567 Factor a perfect square on the left side: (y + -0.611111111)(y + -0.611111111) = 0.0401234567 Calculate the square root of the right side: 0.200308404 Break this problem into two subproblems by setting (y + -0.611111111) equal to 0.200308404 and -0.200308404.Subproblem 1
y + -0.611111111 = 0.200308404 Simplifying y + -0.611111111 = 0.200308404 Reorder the terms: -0.611111111 + y = 0.200308404 Solving -0.611111111 + y = 0.200308404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + y = 0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + y = 0.200308404 + 0.611111111 y = 0.200308404 + 0.611111111 Combine like terms: 0.200308404 + 0.611111111 = 0.811419515 y = 0.811419515 Simplifying y = 0.811419515Subproblem 2
y + -0.611111111 = -0.200308404 Simplifying y + -0.611111111 = -0.200308404 Reorder the terms: -0.611111111 + y = -0.200308404 Solving -0.611111111 + y = -0.200308404 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + y = -0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + y = -0.200308404 + 0.611111111 y = -0.200308404 + 0.611111111 Combine like terms: -0.200308404 + 0.611111111 = 0.410802707 y = 0.410802707 Simplifying y = 0.410802707Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.811419515, 0.410802707}
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