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Simplifying y3 + -16y2 + 36y = 0 Reorder the terms: 36y + -16y2 + y3 = 0 Solving 36y + -16y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(36 + -16y + y2) = 0Subproblem 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 '(36 + -16y + y2)' equal to zero and attempt to solve: Simplifying 36 + -16y + y2 = 0 Solving 36 + -16y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-36' to each side of the equation. 36 + -16y + -36 + y2 = 0 + -36 Reorder the terms: 36 + -36 + -16y + y2 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + -16y + y2 = 0 + -36 -16y + y2 = 0 + -36 Combine like terms: 0 + -36 = -36 -16y + y2 = -36 The y term is -16y. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16y + 64 + y2 = -36 + 64 Reorder the terms: 64 + -16y + y2 = -36 + 64 Combine like terms: -36 + 64 = 28 64 + -16y + y2 = 28 Factor a perfect square on the left side: (y + -8)(y + -8) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (y + -8) equal to 5.291502622 and -5.291502622.Subproblem 1
y + -8 = 5.291502622 Simplifying y + -8 = 5.291502622 Reorder the terms: -8 + y = 5.291502622 Solving -8 + y = 5.291502622 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = 5.291502622 + 8 Combine like terms: -8 + 8 = 0 0 + y = 5.291502622 + 8 y = 5.291502622 + 8 Combine like terms: 5.291502622 + 8 = 13.291502622 y = 13.291502622 Simplifying y = 13.291502622Subproblem 2
y + -8 = -5.291502622 Simplifying y + -8 = -5.291502622 Reorder the terms: -8 + y = -5.291502622 Solving -8 + y = -5.291502622 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = -5.291502622 + 8 Combine like terms: -8 + 8 = 0 0 + y = -5.291502622 + 8 y = -5.291502622 + 8 Combine like terms: -5.291502622 + 8 = 2.708497378 y = 2.708497378 Simplifying y = 2.708497378Solution
The solution to the problem is based on the solutions from the subproblems. y = {13.291502622, 2.708497378}Solution
y = {0, 13.291502622, 2.708497378}
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