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Simplifying 9(3y + -2) * y = 2 Reorder the terms: 9(-2 + 3y) * y = 2 Reorder the terms for easier multiplication: 9y(-2 + 3y) = 2 (-2 * 9y + 3y * 9y) = 2 (-18y + 27y2) = 2 Solving -18y + 27y2 = 2 Solving for variable 'y'. Reorder the terms: -2 + -18y + 27y2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + -18y + 27y2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.07407407407 + -0.6666666667y + y2 = 0 Move the constant term to the right: Add '0.07407407407' to each side of the equation. -0.07407407407 + -0.6666666667y + 0.07407407407 + y2 = 0 + 0.07407407407 Reorder the terms: -0.07407407407 + 0.07407407407 + -0.6666666667y + y2 = 0 + 0.07407407407 Combine like terms: -0.07407407407 + 0.07407407407 = 0.00000000000 0.00000000000 + -0.6666666667y + y2 = 0 + 0.07407407407 -0.6666666667y + y2 = 0 + 0.07407407407 Combine like terms: 0 + 0.07407407407 = 0.07407407407 -0.6666666667y + y2 = 0.07407407407 The y term is -0.6666666667y. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667y + 0.1111111112 + y2 = 0.07407407407 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667y + y2 = 0.07407407407 + 0.1111111112 Combine like terms: 0.07407407407 + 0.1111111112 = 0.18518518527 0.1111111112 + -0.6666666667y + y2 = 0.18518518527 Factor a perfect square on the left side: (y + -0.3333333334)(y + -0.3333333334) = 0.18518518527 Calculate the square root of the right side: 0.430331483 Break this problem into two subproblems by setting (y + -0.3333333334) equal to 0.430331483 and -0.430331483.Subproblem 1
y + -0.3333333334 = 0.430331483 Simplifying y + -0.3333333334 = 0.430331483 Reorder the terms: -0.3333333334 + y = 0.430331483 Solving -0.3333333334 + y = 0.430331483 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = 0.430331483 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 0.430331483 + 0.3333333334 y = 0.430331483 + 0.3333333334 Combine like terms: 0.430331483 + 0.3333333334 = 0.7636648164 y = 0.7636648164 Simplifying y = 0.7636648164Subproblem 2
y + -0.3333333334 = -0.430331483 Simplifying y + -0.3333333334 = -0.430331483 Reorder the terms: -0.3333333334 + y = -0.430331483 Solving -0.3333333334 + y = -0.430331483 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = -0.430331483 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -0.430331483 + 0.3333333334 y = -0.430331483 + 0.3333333334 Combine like terms: -0.430331483 + 0.3333333334 = -0.0969981496 y = -0.0969981496 Simplifying y = -0.0969981496Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.7636648164, -0.0969981496}
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