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Simplifying (3y + 2y2 + y3) + (-6y + -8y2 + 5y3) = 0 Remove parenthesis around (3y + 2y2 + y3) 3y + 2y2 + y3 + (-6y + -8y2 + 5y3) = 0 Remove parenthesis around (-6y + -8y2 + 5y3) 3y + 2y2 + y3 + -6y + -8y2 + 5y3 = 0 Reorder the terms: 3y + -6y + 2y2 + -8y2 + y3 + 5y3 = 0 Combine like terms: 3y + -6y = -3y -3y + 2y2 + -8y2 + y3 + 5y3 = 0 Combine like terms: 2y2 + -8y2 = -6y2 -3y + -6y2 + y3 + 5y3 = 0 Combine like terms: y3 + 5y3 = 6y3 -3y + -6y2 + 6y3 = 0 Solving -3y + -6y2 + 6y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3y'. 3y(-1 + -2y + 2y2) = 0 Ignore the factor 3.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 + 2y2)' equal to zero and attempt to solve: Simplifying -1 + -2y + 2y2 = 0 Solving -1 + -2y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -1y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1y + y2 = 0 + 0.5 -1y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1y + y2 = 0.5 The y term is -1y. 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. -1y + 0.25 + y2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1y + y2 = 0.75 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (y + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
y + -0.5 = 0.866025404 Simplifying y + -0.5 = 0.866025404 Reorder the terms: -0.5 + y = 0.866025404 Solving -0.5 + y = 0.866025404 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 = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 0.866025404 + 0.5 y = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 y = 1.366025404 Simplifying y = 1.366025404Subproblem 2
y + -0.5 = -0.866025404 Simplifying y + -0.5 = -0.866025404 Reorder the terms: -0.5 + y = -0.866025404 Solving -0.5 + y = -0.866025404 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 = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -0.866025404 + 0.5 y = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 y = -0.366025404 Simplifying y = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.366025404, -0.366025404}Solution
y = {0, 1.366025404, -0.366025404}
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