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Simplifying 9y2 + 22 = 30y Reorder the terms: 22 + 9y2 = 30y Solving 22 + 9y2 = 30y Solving for variable 'y'. Reorder the terms: 22 + -30y + 9y2 = 30y + -30y Combine like terms: 30y + -30y = 0 22 + -30y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.444444444 + -3.333333333y + y2 = 0 Move the constant term to the right: Add '-2.444444444' to each side of the equation. 2.444444444 + -3.333333333y + -2.444444444 + y2 = 0 + -2.444444444 Reorder the terms: 2.444444444 + -2.444444444 + -3.333333333y + y2 = 0 + -2.444444444 Combine like terms: 2.444444444 + -2.444444444 = 0.000000000 0.000000000 + -3.333333333y + y2 = 0 + -2.444444444 -3.333333333y + y2 = 0 + -2.444444444 Combine like terms: 0 + -2.444444444 = -2.444444444 -3.333333333y + y2 = -2.444444444 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 = -2.444444444 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333y + y2 = -2.444444444 + 2.777777779 Combine like terms: -2.444444444 + 2.777777779 = 0.333333335 2.777777779 + -3.333333333y + y2 = 0.333333335 Factor a perfect square on the left side: (y + -1.666666667)(y + -1.666666667) = 0.333333335 Calculate the square root of the right side: 0.577350271 Break this problem into two subproblems by setting (y + -1.666666667) equal to 0.577350271 and -0.577350271.Subproblem 1
y + -1.666666667 = 0.577350271 Simplifying y + -1.666666667 = 0.577350271 Reorder the terms: -1.666666667 + y = 0.577350271 Solving -1.666666667 + y = 0.577350271 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 = 0.577350271 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 0.577350271 + 1.666666667 y = 0.577350271 + 1.666666667 Combine like terms: 0.577350271 + 1.666666667 = 2.244016938 y = 2.244016938 Simplifying y = 2.244016938Subproblem 2
y + -1.666666667 = -0.577350271 Simplifying y + -1.666666667 = -0.577350271 Reorder the terms: -1.666666667 + y = -0.577350271 Solving -1.666666667 + y = -0.577350271 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 = -0.577350271 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -0.577350271 + 1.666666667 y = -0.577350271 + 1.666666667 Combine like terms: -0.577350271 + 1.666666667 = 1.089316396 y = 1.089316396 Simplifying y = 1.089316396Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.244016938, 1.089316396}
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