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Simplifying 14y + 22 + -15y2 + 23y = 0 Reorder the terms: 22 + 14y + 23y + -15y2 = 0 Combine like terms: 14y + 23y = 37y 22 + 37y + -15y2 = 0 Solving 22 + 37y + -15y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -1.466666667 + -2.466666667y + y2 = 0 Move the constant term to the right: Add '1.466666667' to each side of the equation. -1.466666667 + -2.466666667y + 1.466666667 + y2 = 0 + 1.466666667 Reorder the terms: -1.466666667 + 1.466666667 + -2.466666667y + y2 = 0 + 1.466666667 Combine like terms: -1.466666667 + 1.466666667 = 0.000000000 0.000000000 + -2.466666667y + y2 = 0 + 1.466666667 -2.466666667y + y2 = 0 + 1.466666667 Combine like terms: 0 + 1.466666667 = 1.466666667 -2.466666667y + y2 = 1.466666667 The y term is -2.466666667y. Take half its coefficient (-1.233333334). Square it (1.521111113) and add it to both sides. Add '1.521111113' to each side of the equation. -2.466666667y + 1.521111113 + y2 = 1.466666667 + 1.521111113 Reorder the terms: 1.521111113 + -2.466666667y + y2 = 1.466666667 + 1.521111113 Combine like terms: 1.466666667 + 1.521111113 = 2.98777778 1.521111113 + -2.466666667y + y2 = 2.98777778 Factor a perfect square on the left side: (y + -1.233333334)(y + -1.233333334) = 2.98777778 Calculate the square root of the right side: 1.728518956 Break this problem into two subproblems by setting (y + -1.233333334) equal to 1.728518956 and -1.728518956.Subproblem 1
y + -1.233333334 = 1.728518956 Simplifying y + -1.233333334 = 1.728518956 Reorder the terms: -1.233333334 + y = 1.728518956 Solving -1.233333334 + y = 1.728518956 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + y = 1.728518956 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + y = 1.728518956 + 1.233333334 y = 1.728518956 + 1.233333334 Combine like terms: 1.728518956 + 1.233333334 = 2.96185229 y = 2.96185229 Simplifying y = 2.96185229Subproblem 2
y + -1.233333334 = -1.728518956 Simplifying y + -1.233333334 = -1.728518956 Reorder the terms: -1.233333334 + y = -1.728518956 Solving -1.233333334 + y = -1.728518956 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + y = -1.728518956 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + y = -1.728518956 + 1.233333334 y = -1.728518956 + 1.233333334 Combine like terms: -1.728518956 + 1.233333334 = -0.495185622 y = -0.495185622 Simplifying y = -0.495185622Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.96185229, -0.495185622}
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