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Simplifying 9y2 + -24y + 10 = 0 Reorder the terms: 10 + -24y + 9y2 = 0 Solving 10 + -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'. 1.111111111 + -2.666666667y + y2 = 0 Move the constant term to the right: Add '-1.111111111' to each side of the equation. 1.111111111 + -2.666666667y + -1.111111111 + y2 = 0 + -1.111111111 Reorder the terms: 1.111111111 + -1.111111111 + -2.666666667y + y2 = 0 + -1.111111111 Combine like terms: 1.111111111 + -1.111111111 = 0.000000000 0.000000000 + -2.666666667y + y2 = 0 + -1.111111111 -2.666666667y + y2 = 0 + -1.111111111 Combine like terms: 0 + -1.111111111 = -1.111111111 -2.666666667y + y2 = -1.111111111 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 = -1.111111111 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667y + y2 = -1.111111111 + 1.777777780 Combine like terms: -1.111111111 + 1.777777780 = 0.666666669 1.777777780 + -2.666666667y + y2 = 0.666666669 Factor a perfect square on the left side: (y + -1.333333334)(y + -1.333333334) = 0.666666669 Calculate the square root of the right side: 0.816496582 Break this problem into two subproblems by setting (y + -1.333333334) equal to 0.816496582 and -0.816496582.Subproblem 1
y + -1.333333334 = 0.816496582 Simplifying y + -1.333333334 = 0.816496582 Reorder the terms: -1.333333334 + y = 0.816496582 Solving -1.333333334 + y = 0.816496582 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 = 0.816496582 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 0.816496582 + 1.333333334 y = 0.816496582 + 1.333333334 Combine like terms: 0.816496582 + 1.333333334 = 2.149829916 y = 2.149829916 Simplifying y = 2.149829916Subproblem 2
y + -1.333333334 = -0.816496582 Simplifying y + -1.333333334 = -0.816496582 Reorder the terms: -1.333333334 + y = -0.816496582 Solving -1.333333334 + y = -0.816496582 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 = -0.816496582 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -0.816496582 + 1.333333334 y = -0.816496582 + 1.333333334 Combine like terms: -0.816496582 + 1.333333334 = 0.516836752 y = 0.516836752 Simplifying y = 0.516836752Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.149829916, 0.516836752}
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