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Simplifying y3 + -13y2 + 3y = 0 Reorder the terms: 3y + -13y2 + y3 = 0 Solving 3y + -13y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(3 + -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 '(3 + -13y + y2)' equal to zero and attempt to solve: Simplifying 3 + -13y + y2 = 0 Solving 3 + -13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -13y + -3 + y2 = 0 + -3 Reorder the terms: 3 + -3 + -13y + y2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -13y + y2 = 0 + -3 -13y + y2 = 0 + -3 Combine like terms: 0 + -3 = -3 -13y + y2 = -3 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 = -3 + 42.25 Reorder the terms: 42.25 + -13y + y2 = -3 + 42.25 Combine like terms: -3 + 42.25 = 39.25 42.25 + -13y + y2 = 39.25 Factor a perfect square on the left side: (y + -6.5)(y + -6.5) = 39.25 Calculate the square root of the right side: 6.264982043 Break this problem into two subproblems by setting (y + -6.5) equal to 6.264982043 and -6.264982043.Subproblem 1
y + -6.5 = 6.264982043 Simplifying y + -6.5 = 6.264982043 Reorder the terms: -6.5 + y = 6.264982043 Solving -6.5 + y = 6.264982043 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 = 6.264982043 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 6.264982043 + 6.5 y = 6.264982043 + 6.5 Combine like terms: 6.264982043 + 6.5 = 12.764982043 y = 12.764982043 Simplifying y = 12.764982043Subproblem 2
y + -6.5 = -6.264982043 Simplifying y + -6.5 = -6.264982043 Reorder the terms: -6.5 + y = -6.264982043 Solving -6.5 + y = -6.264982043 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 = -6.264982043 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -6.264982043 + 6.5 y = -6.264982043 + 6.5 Combine like terms: -6.264982043 + 6.5 = 0.235017957 y = 0.235017957 Simplifying y = 0.235017957Solution
The solution to the problem is based on the solutions from the subproblems. y = {12.764982043, 0.235017957}Solution
y = {0, 12.764982043, 0.235017957}
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