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Simplifying 9y2 + 13 = 24y Reorder the terms: 13 + 9y2 = 24y Solving 13 + 9y2 = 24y Solving for variable 'y'. Reorder the terms: 13 + -24y + 9y2 = 24y + -24y Combine like terms: 24y + -24y = 0 13 + -24y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.444444444 + -2.666666667y + y2 = 0 Move the constant term to the right: Add '-1.444444444' to each side of the equation. 1.444444444 + -2.666666667y + -1.444444444 + y2 = 0 + -1.444444444 Reorder the terms: 1.444444444 + -1.444444444 + -2.666666667y + y2 = 0 + -1.444444444 Combine like terms: 1.444444444 + -1.444444444 = 0.000000000 0.000000000 + -2.666666667y + y2 = 0 + -1.444444444 -2.666666667y + y2 = 0 + -1.444444444 Combine like terms: 0 + -1.444444444 = -1.444444444 -2.666666667y + y2 = -1.444444444 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 = -1.444444444 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667y + y2 = -1.444444444 + 1.777777780 Combine like terms: -1.444444444 + 1.777777780 = 0.333333336 1.777777780 + -2.666666667y + y2 = 0.333333336 Factor a perfect square on the left side: (y + -1.333333334)(y + -1.333333334) = 0.333333336 Calculate the square root of the right side: 0.577350271 Break this problem into two subproblems by setting (y + -1.333333334) equal to 0.577350271 and -0.577350271.Subproblem 1
y + -1.333333334 = 0.577350271 Simplifying y + -1.333333334 = 0.577350271 Reorder the terms: -1.333333334 + y = 0.577350271 Solving -1.333333334 + y = 0.577350271 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 = 0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 0.577350271 + 1.333333334 y = 0.577350271 + 1.333333334 Combine like terms: 0.577350271 + 1.333333334 = 1.910683605 y = 1.910683605 Simplifying y = 1.910683605Subproblem 2
y + -1.333333334 = -0.577350271 Simplifying y + -1.333333334 = -0.577350271 Reorder the terms: -1.333333334 + y = -0.577350271 Solving -1.333333334 + y = -0.577350271 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 = -0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -0.577350271 + 1.333333334 y = -0.577350271 + 1.333333334 Combine like terms: -0.577350271 + 1.333333334 = 0.755983063 y = 0.755983063 Simplifying y = 0.755983063Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.910683605, 0.755983063}
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