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Simplifying 27y3 + 18y2 + -81y = 0 Reorder the terms: -81y + 18y2 + 27y3 = 0 Solving -81y + 18y2 + 27y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '9y'. 9y(-9 + 2y + 3y2) = 0 Ignore the factor 9.Subproblem 1
Set the factor 'y' equal to zero and attempt to solve: Simplifying y = 0 Solving y = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y = 0Subproblem 2
Set the factor '(-9 + 2y + 3y2)' equal to zero and attempt to solve: Simplifying -9 + 2y + 3y2 = 0 Solving -9 + 2y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3 + 0.6666666667y + y2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.6666666667y + 3 + y2 = 0 + 3 Reorder the terms: -3 + 3 + 0.6666666667y + y2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.6666666667y + y2 = 0 + 3 0.6666666667y + y2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.6666666667y + y2 = 3 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 = 3 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667y + y2 = 3 + 0.1111111112 Combine like terms: 3 + 0.1111111112 = 3.1111111112 0.1111111112 + 0.6666666667y + y2 = 3.1111111112 Factor a perfect square on the left side: (y + 0.3333333334)(y + 0.3333333334) = 3.1111111112 Calculate the square root of the right side: 1.763834207 Break this problem into two subproblems by setting (y + 0.3333333334) equal to 1.763834207 and -1.763834207.Subproblem 1
y + 0.3333333334 = 1.763834207 Simplifying y + 0.3333333334 = 1.763834207 Reorder the terms: 0.3333333334 + y = 1.763834207 Solving 0.3333333334 + y = 1.763834207 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 = 1.763834207 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = 1.763834207 + -0.3333333334 y = 1.763834207 + -0.3333333334 Combine like terms: 1.763834207 + -0.3333333334 = 1.4305008736 y = 1.4305008736 Simplifying y = 1.4305008736Subproblem 2
y + 0.3333333334 = -1.763834207 Simplifying y + 0.3333333334 = -1.763834207 Reorder the terms: 0.3333333334 + y = -1.763834207 Solving 0.3333333334 + y = -1.763834207 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 = -1.763834207 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + y = -1.763834207 + -0.3333333334 y = -1.763834207 + -0.3333333334 Combine like terms: -1.763834207 + -0.3333333334 = -2.0971675404 y = -2.0971675404 Simplifying y = -2.0971675404Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.4305008736, -2.0971675404}Solution
y = {0, 1.4305008736, -2.0971675404}
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