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Simplifying 5y3 + -5y2 + -5y2 + -5y = 0 Reorder the terms: -5y + -5y2 + -5y2 + 5y3 = 0 Combine like terms: -5y2 + -5y2 = -10y2 -5y + -10y2 + 5y3 = 0 Solving -5y + -10y2 + 5y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '5y'. 5y(-1 + -2y + y2) = 0 Ignore the factor 5.Subproblem 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 '(-1 + -2y + y2)' equal to zero and attempt to solve: Simplifying -1 + -2y + y2 = 0 Solving -1 + -2y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -2y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + -2y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2y + y2 = 0 + 1 -2y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2y + y2 = 1 The y term is -2y. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2y + 1 + y2 = 1 + 1 Reorder the terms: 1 + -2y + y2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2y + y2 = 2 Factor a perfect square on the left side: (y + -1)(y + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (y + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
y + -1 = 1.414213562 Simplifying y + -1 = 1.414213562 Reorder the terms: -1 + y = 1.414213562 Solving -1 + y = 1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + y = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + y = 1.414213562 + 1 y = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 y = 2.414213562 Simplifying y = 2.414213562Subproblem 2
y + -1 = -1.414213562 Simplifying y + -1 = -1.414213562 Reorder the terms: -1 + y = -1.414213562 Solving -1 + y = -1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + y = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + y = -1.414213562 + 1 y = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 y = -0.414213562 Simplifying y = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.414213562, -0.414213562}Solution
y = {0, 2.414213562, -0.414213562}
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