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Simplifying 9y2 + 23 = 30y Reorder the terms: 23 + 9y2 = 30y Solving 23 + 9y2 = 30y Solving for variable 'y'. Reorder the terms: 23 + -30y + 9y2 = 30y + -30y Combine like terms: 30y + -30y = 0 23 + -30y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.555555556 + -3.333333333y + y2 = 0 Move the constant term to the right: Add '-2.555555556' to each side of the equation. 2.555555556 + -3.333333333y + -2.555555556 + y2 = 0 + -2.555555556 Reorder the terms: 2.555555556 + -2.555555556 + -3.333333333y + y2 = 0 + -2.555555556 Combine like terms: 2.555555556 + -2.555555556 = 0.000000000 0.000000000 + -3.333333333y + y2 = 0 + -2.555555556 -3.333333333y + y2 = 0 + -2.555555556 Combine like terms: 0 + -2.555555556 = -2.555555556 -3.333333333y + y2 = -2.555555556 The y term is -3.333333333y. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333y + 2.777777779 + y2 = -2.555555556 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333y + y2 = -2.555555556 + 2.777777779 Combine like terms: -2.555555556 + 2.777777779 = 0.222222223 2.777777779 + -3.333333333y + y2 = 0.222222223 Factor a perfect square on the left side: (y + -1.666666667)(y + -1.666666667) = 0.222222223 Calculate the square root of the right side: 0.471404522 Break this problem into two subproblems by setting (y + -1.666666667) equal to 0.471404522 and -0.471404522.Subproblem 1
y + -1.666666667 = 0.471404522 Simplifying y + -1.666666667 = 0.471404522 Reorder the terms: -1.666666667 + y = 0.471404522 Solving -1.666666667 + y = 0.471404522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = 0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 0.471404522 + 1.666666667 y = 0.471404522 + 1.666666667 Combine like terms: 0.471404522 + 1.666666667 = 2.138071189 y = 2.138071189 Simplifying y = 2.138071189Subproblem 2
y + -1.666666667 = -0.471404522 Simplifying y + -1.666666667 = -0.471404522 Reorder the terms: -1.666666667 + y = -0.471404522 Solving -1.666666667 + y = -0.471404522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = -0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -0.471404522 + 1.666666667 y = -0.471404522 + 1.666666667 Combine like terms: -0.471404522 + 1.666666667 = 1.195262145 y = 1.195262145 Simplifying y = 1.195262145Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.138071189, 1.195262145}
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