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Simplifying 9y2 + -30y = -20 Reorder the terms: -30y + 9y2 = -20 Solving -30y + 9y2 = -20 Solving for variable 'y'. Reorder the terms: 20 + -30y + 9y2 = -20 + 20 Combine like terms: -20 + 20 = 0 20 + -30y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.222222222 + -3.333333333y + y2 = 0 Move the constant term to the right: Add '-2.222222222' to each side of the equation. 2.222222222 + -3.333333333y + -2.222222222 + y2 = 0 + -2.222222222 Reorder the terms: 2.222222222 + -2.222222222 + -3.333333333y + y2 = 0 + -2.222222222 Combine like terms: 2.222222222 + -2.222222222 = 0.000000000 0.000000000 + -3.333333333y + y2 = 0 + -2.222222222 -3.333333333y + y2 = 0 + -2.222222222 Combine like terms: 0 + -2.222222222 = -2.222222222 -3.333333333y + y2 = -2.222222222 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 = -2.222222222 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333y + y2 = -2.222222222 + 2.777777779 Combine like terms: -2.222222222 + 2.777777779 = 0.555555557 2.777777779 + -3.333333333y + y2 = 0.555555557 Factor a perfect square on the left side: (y + -1.666666667)(y + -1.666666667) = 0.555555557 Calculate the square root of the right side: 0.745355993 Break this problem into two subproblems by setting (y + -1.666666667) equal to 0.745355993 and -0.745355993.Subproblem 1
y + -1.666666667 = 0.745355993 Simplifying y + -1.666666667 = 0.745355993 Reorder the terms: -1.666666667 + y = 0.745355993 Solving -1.666666667 + y = 0.745355993 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 = 0.745355993 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 0.745355993 + 1.666666667 y = 0.745355993 + 1.666666667 Combine like terms: 0.745355993 + 1.666666667 = 2.41202266 y = 2.41202266 Simplifying y = 2.41202266Subproblem 2
y + -1.666666667 = -0.745355993 Simplifying y + -1.666666667 = -0.745355993 Reorder the terms: -1.666666667 + y = -0.745355993 Solving -1.666666667 + y = -0.745355993 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 = -0.745355993 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -0.745355993 + 1.666666667 y = -0.745355993 + 1.666666667 Combine like terms: -0.745355993 + 1.666666667 = 0.921310674 y = 0.921310674 Simplifying y = 0.921310674Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.41202266, 0.921310674}
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