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Simplifying 12y3 + 6y2 + -3y = 0 Reorder the terms: -3y + 6y2 + 12y3 = 0 Solving -3y + 6y2 + 12y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3y'. 3y(-1 + 2y + 4y2) = 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 + 4y2)' equal to zero and attempt to solve: Simplifying -1 + 2y + 4y2 = 0 Solving -1 + 2y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 0.5y + y2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.5y + 0.25 + y2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.5y + y2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.5y + y2 = 0 + 0.25 0.5y + y2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.5y + y2 = 0.25 The y term is 0.5y. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5y + 0.0625 + y2 = 0.25 + 0.0625 Reorder the terms: 0.0625 + 0.5y + y2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + 0.5y + y2 = 0.3125 Factor a perfect square on the left side: (y + 0.25)(y + 0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (y + 0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
y + 0.25 = 0.559016994 Simplifying y + 0.25 = 0.559016994 Reorder the terms: 0.25 + y = 0.559016994 Solving 0.25 + y = 0.559016994 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = 0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = 0.559016994 + -0.25 y = 0.559016994 + -0.25 Combine like terms: 0.559016994 + -0.25 = 0.309016994 y = 0.309016994 Simplifying y = 0.309016994Subproblem 2
y + 0.25 = -0.559016994 Simplifying y + 0.25 = -0.559016994 Reorder the terms: 0.25 + y = -0.559016994 Solving 0.25 + y = -0.559016994 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = -0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = -0.559016994 + -0.25 y = -0.559016994 + -0.25 Combine like terms: -0.559016994 + -0.25 = -0.809016994 y = -0.809016994 Simplifying y = -0.809016994Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.309016994, -0.809016994}Solution
y = {0, 0.309016994, -0.809016994}
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