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Simplifying 9y2 + -24y + 8 = 0 Reorder the terms: 8 + -24y + 9y2 = 0 Solving 8 + -24y + 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'. 0.8888888889 + -2.666666667y + y2 = 0 Move the constant term to the right: Add '-0.8888888889' to each side of the equation. 0.8888888889 + -2.666666667y + -0.8888888889 + y2 = 0 + -0.8888888889 Reorder the terms: 0.8888888889 + -0.8888888889 + -2.666666667y + y2 = 0 + -0.8888888889 Combine like terms: 0.8888888889 + -0.8888888889 = 0.0000000000 0.0000000000 + -2.666666667y + y2 = 0 + -0.8888888889 -2.666666667y + y2 = 0 + -0.8888888889 Combine like terms: 0 + -0.8888888889 = -0.8888888889 -2.666666667y + y2 = -0.8888888889 The y term is -2.666666667y. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667y + 1.777777780 + y2 = -0.8888888889 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667y + y2 = -0.8888888889 + 1.777777780 Combine like terms: -0.8888888889 + 1.777777780 = 0.8888888911 1.777777780 + -2.666666667y + y2 = 0.8888888911 Factor a perfect square on the left side: (y + -1.333333334)(y + -1.333333334) = 0.8888888911 Calculate the square root of the right side: 0.942809043 Break this problem into two subproblems by setting (y + -1.333333334) equal to 0.942809043 and -0.942809043.Subproblem 1
y + -1.333333334 = 0.942809043 Simplifying y + -1.333333334 = 0.942809043 Reorder the terms: -1.333333334 + y = 0.942809043 Solving -1.333333334 + y = 0.942809043 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 0.942809043 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 0.942809043 + 1.333333334 y = 0.942809043 + 1.333333334 Combine like terms: 0.942809043 + 1.333333334 = 2.276142377 y = 2.276142377 Simplifying y = 2.276142377Subproblem 2
y + -1.333333334 = -0.942809043 Simplifying y + -1.333333334 = -0.942809043 Reorder the terms: -1.333333334 + y = -0.942809043 Solving -1.333333334 + y = -0.942809043 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -0.942809043 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -0.942809043 + 1.333333334 y = -0.942809043 + 1.333333334 Combine like terms: -0.942809043 + 1.333333334 = 0.390524291 y = 0.390524291 Simplifying y = 0.390524291Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.276142377, 0.390524291}
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