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Simplifying 15y3 + 20y2 + -5y = 0 Reorder the terms: -5y + 20y2 + 15y3 = 0 Solving -5y + 20y2 + 15y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '5y'. 5y(-1 + 4y + 3y2) = 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 + 4y + 3y2)' equal to zero and attempt to solve: Simplifying -1 + 4y + 3y2 = 0 Solving -1 + 4y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + 1.333333333y + y2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 1.333333333y + 0.3333333333 + y2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 1.333333333y + y2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 1.333333333y + y2 = 0 + 0.3333333333 1.333333333y + y2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 1.333333333y + y2 = 0.3333333333 The y term is 1.333333333y. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333y + 0.4444444442 + y2 = 0.3333333333 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333y + y2 = 0.3333333333 + 0.4444444442 Combine like terms: 0.3333333333 + 0.4444444442 = 0.7777777775 0.4444444442 + 1.333333333y + y2 = 0.7777777775 Factor a perfect square on the left side: (y + 0.6666666665)(y + 0.6666666665) = 0.7777777775 Calculate the square root of the right side: 0.881917104 Break this problem into two subproblems by setting (y + 0.6666666665) equal to 0.881917104 and -0.881917104.Subproblem 1
y + 0.6666666665 = 0.881917104 Simplifying y + 0.6666666665 = 0.881917104 Reorder the terms: 0.6666666665 + y = 0.881917104 Solving 0.6666666665 + y = 0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = 0.881917104 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = 0.881917104 + -0.6666666665 y = 0.881917104 + -0.6666666665 Combine like terms: 0.881917104 + -0.6666666665 = 0.2152504375 y = 0.2152504375 Simplifying y = 0.2152504375Subproblem 2
y + 0.6666666665 = -0.881917104 Simplifying y + 0.6666666665 = -0.881917104 Reorder the terms: 0.6666666665 + y = -0.881917104 Solving 0.6666666665 + y = -0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = -0.881917104 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = -0.881917104 + -0.6666666665 y = -0.881917104 + -0.6666666665 Combine like terms: -0.881917104 + -0.6666666665 = -1.5485837705 y = -1.5485837705 Simplifying y = -1.5485837705Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.2152504375, -1.5485837705}Solution
y = {0, 0.2152504375, -1.5485837705}
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