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Simplifying 81y2 + -45y + 9y3 = 0 Reorder the terms: -45y + 81y2 + 9y3 = 0 Solving -45y + 81y2 + 9y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '9y'. 9y(-5 + 9y + y2) = 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 '(-5 + 9y + y2)' equal to zero and attempt to solve: Simplifying -5 + 9y + y2 = 0 Solving -5 + 9y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 9y + 5 + y2 = 0 + 5 Reorder the terms: -5 + 5 + 9y + y2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 9y + y2 = 0 + 5 9y + y2 = 0 + 5 Combine like terms: 0 + 5 = 5 9y + y2 = 5 The y term is 9y. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9y + 20.25 + y2 = 5 + 20.25 Reorder the terms: 20.25 + 9y + y2 = 5 + 20.25 Combine like terms: 5 + 20.25 = 25.25 20.25 + 9y + y2 = 25.25 Factor a perfect square on the left side: (y + 4.5)(y + 4.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (y + 4.5) equal to 5.024937811 and -5.024937811.Subproblem 1
y + 4.5 = 5.024937811 Simplifying y + 4.5 = 5.024937811 Reorder the terms: 4.5 + y = 5.024937811 Solving 4.5 + y = 5.024937811 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + y = 5.024937811 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + y = 5.024937811 + -4.5 y = 5.024937811 + -4.5 Combine like terms: 5.024937811 + -4.5 = 0.524937811 y = 0.524937811 Simplifying y = 0.524937811Subproblem 2
y + 4.5 = -5.024937811 Simplifying y + 4.5 = -5.024937811 Reorder the terms: 4.5 + y = -5.024937811 Solving 4.5 + y = -5.024937811 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + y = -5.024937811 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + y = -5.024937811 + -4.5 y = -5.024937811 + -4.5 Combine like terms: -5.024937811 + -4.5 = -9.524937811 y = -9.524937811 Simplifying y = -9.524937811Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.524937811, -9.524937811}Solution
y = {0, 0.524937811, -9.524937811}
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