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Simplifying y4 + -13y3 = 42y2 Reorder the terms: -13y3 + y4 = 42y2 Solving -13y3 + y4 = 42y2 Solving for variable 'y'. Reorder the terms: -42y2 + -13y3 + y4 = 42y2 + -42y2 Combine like terms: 42y2 + -42y2 = 0 -42y2 + -13y3 + y4 = 0 Factor out the Greatest Common Factor (GCF), 'y2'. y2(-42 + -13y + y2) = 0Subproblem 1
Set the factor 'y2' equal to zero and attempt to solve: Simplifying y2 = 0 Solving y2 = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y2 = 0 Take the square root of each side: y = {0}Subproblem 2
Set the factor '(-42 + -13y + y2)' equal to zero and attempt to solve: Simplifying -42 + -13y + y2 = 0 Solving -42 + -13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '42' to each side of the equation. -42 + -13y + 42 + y2 = 0 + 42 Reorder the terms: -42 + 42 + -13y + y2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + -13y + y2 = 0 + 42 -13y + y2 = 0 + 42 Combine like terms: 0 + 42 = 42 -13y + y2 = 42 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 = 42 + 42.25 Reorder the terms: 42.25 + -13y + y2 = 42 + 42.25 Combine like terms: 42 + 42.25 = 84.25 42.25 + -13y + y2 = 84.25 Factor a perfect square on the left side: (y + -6.5)(y + -6.5) = 84.25 Calculate the square root of the right side: 9.178779875 Break this problem into two subproblems by setting (y + -6.5) equal to 9.178779875 and -9.178779875.Subproblem 1
y + -6.5 = 9.178779875 Simplifying y + -6.5 = 9.178779875 Reorder the terms: -6.5 + y = 9.178779875 Solving -6.5 + y = 9.178779875 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 = 9.178779875 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 9.178779875 + 6.5 y = 9.178779875 + 6.5 Combine like terms: 9.178779875 + 6.5 = 15.678779875 y = 15.678779875 Simplifying y = 15.678779875Subproblem 2
y + -6.5 = -9.178779875 Simplifying y + -6.5 = -9.178779875 Reorder the terms: -6.5 + y = -9.178779875 Solving -6.5 + y = -9.178779875 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 = -9.178779875 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -9.178779875 + 6.5 y = -9.178779875 + 6.5 Combine like terms: -9.178779875 + 6.5 = -2.678779875 y = -2.678779875 Simplifying y = -2.678779875Solution
The solution to the problem is based on the solutions from the subproblems. y = {15.678779875, -2.678779875}Solution
y = {0, 15.678779875, -2.678779875}
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