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Simplifying -1y3 + -1y2 + y = 0 Reorder the terms: y + -1y2 + -1y3 = 0 Solving y + -1y2 + -1y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(1 + -1y + -1y2) = 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 '(1 + -1y + -1y2)' equal to zero and attempt to solve: Simplifying 1 + -1y + -1y2 = 0 Solving 1 + -1y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + y + y2 = 0 + 1 y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 y + y2 = 1 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + y2 = 1 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 1.25 + y2 = 1.25 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (y + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
y + 0.5 = 1.118033989 Simplifying y + 0.5 = 1.118033989 Reorder the terms: 0.5 + y = 1.118033989 Solving 0.5 + y = 1.118033989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.118033989 + -0.5 y = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 y = 0.618033989 Simplifying y = 0.618033989Subproblem 2
y + 0.5 = -1.118033989 Simplifying y + 0.5 = -1.118033989 Reorder the terms: 0.5 + y = -1.118033989 Solving 0.5 + y = -1.118033989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.118033989 + -0.5 y = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 y = -1.618033989 Simplifying y = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.618033989, -1.618033989}Solution
y = {0, 0.618033989, -1.618033989}
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