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Simplifying y3 + -13y2 + -36y = 0 Reorder the terms: -36y + -13y2 + y3 = 0 Solving -36y + -13y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-36 + -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 '(-36 + -13y + y2)' equal to zero and attempt to solve: Simplifying -36 + -13y + y2 = 0 Solving -36 + -13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '36' to each side of the equation. -36 + -13y + 36 + y2 = 0 + 36 Reorder the terms: -36 + 36 + -13y + y2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + -13y + y2 = 0 + 36 -13y + y2 = 0 + 36 Combine like terms: 0 + 36 = 36 -13y + y2 = 36 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 = 36 + 42.25 Reorder the terms: 42.25 + -13y + y2 = 36 + 42.25 Combine like terms: 36 + 42.25 = 78.25 42.25 + -13y + y2 = 78.25 Factor a perfect square on the left side: (y + -6.5)(y + -6.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (y + -6.5) equal to 8.845903006 and -8.845903006.Subproblem 1
y + -6.5 = 8.845903006 Simplifying y + -6.5 = 8.845903006 Reorder the terms: -6.5 + y = 8.845903006 Solving -6.5 + y = 8.845903006 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 = 8.845903006 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 8.845903006 + 6.5 y = 8.845903006 + 6.5 Combine like terms: 8.845903006 + 6.5 = 15.345903006 y = 15.345903006 Simplifying y = 15.345903006Subproblem 2
y + -6.5 = -8.845903006 Simplifying y + -6.5 = -8.845903006 Reorder the terms: -6.5 + y = -8.845903006 Solving -6.5 + y = -8.845903006 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 = -8.845903006 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -8.845903006 + 6.5 y = -8.845903006 + 6.5 Combine like terms: -8.845903006 + 6.5 = -2.345903006 y = -2.345903006 Simplifying y = -2.345903006Solution
The solution to the problem is based on the solutions from the subproblems. y = {15.345903006, -2.345903006}Solution
y = {0, 15.345903006, -2.345903006}
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