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Simplifying y3 + -13y2 + -40y = 0 Reorder the terms: -40y + -13y2 + y3 = 0 Solving -40y + -13y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-40 + -13y + 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 '(-40 + -13y + y2)' equal to zero and attempt to solve: Simplifying -40 + -13y + y2 = 0 Solving -40 + -13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + -13y + 40 + y2 = 0 + 40 Reorder the terms: -40 + 40 + -13y + y2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -13y + y2 = 0 + 40 -13y + y2 = 0 + 40 Combine like terms: 0 + 40 = 40 -13y + y2 = 40 The y term is -13y. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13y + 42.25 + y2 = 40 + 42.25 Reorder the terms: 42.25 + -13y + y2 = 40 + 42.25 Combine like terms: 40 + 42.25 = 82.25 42.25 + -13y + y2 = 82.25 Factor a perfect square on the left side: (y + -6.5)(y + -6.5) = 82.25 Calculate the square root of the right side: 9.069178574 Break this problem into two subproblems by setting (y + -6.5) equal to 9.069178574 and -9.069178574.Subproblem 1
y + -6.5 = 9.069178574 Simplifying y + -6.5 = 9.069178574 Reorder the terms: -6.5 + y = 9.069178574 Solving -6.5 + y = 9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = 9.069178574 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 9.069178574 + 6.5 y = 9.069178574 + 6.5 Combine like terms: 9.069178574 + 6.5 = 15.569178574 y = 15.569178574 Simplifying y = 15.569178574Subproblem 2
y + -6.5 = -9.069178574 Simplifying y + -6.5 = -9.069178574 Reorder the terms: -6.5 + y = -9.069178574 Solving -6.5 + y = -9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = -9.069178574 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -9.069178574 + 6.5 y = -9.069178574 + 6.5 Combine like terms: -9.069178574 + 6.5 = -2.569178574 y = -2.569178574 Simplifying y = -2.569178574Solution
The solution to the problem is based on the solutions from the subproblems. y = {15.569178574, -2.569178574}Solution
y = {0, 15.569178574, -2.569178574}
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