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Simplifying 9y4 + -15y3 + 3y2 = 0 Reorder the terms: 3y2 + -15y3 + 9y4 = 0 Solving 3y2 + -15y3 + 9y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3y2'. 3y2(1 + -5y + 3y2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'y2' equal to zero and attempt to solve: Simplifying y2 = 0 Solving y2 = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y2 = 0 Take the square root of each side: y = {0}Subproblem 2
Set the factor '(1 + -5y + 3y2)' equal to zero and attempt to solve: Simplifying 1 + -5y + 3y2 = 0 Solving 1 + -5y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + -1.666666667y + y2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -1.666666667y + -0.3333333333 + y2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -1.666666667y + y2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -1.666666667y + y2 = 0 + -0.3333333333 -1.666666667y + y2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -1.666666667y + y2 = -0.3333333333 The y term is -1.666666667y. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667y + 0.6944444447 + y2 = -0.3333333333 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667y + y2 = -0.3333333333 + 0.6944444447 Combine like terms: -0.3333333333 + 0.6944444447 = 0.3611111114 0.6944444447 + -1.666666667y + y2 = 0.3611111114 Factor a perfect square on the left side: (y + -0.8333333335)(y + -0.8333333335) = 0.3611111114 Calculate the square root of the right side: 0.600925213 Break this problem into two subproblems by setting (y + -0.8333333335) equal to 0.600925213 and -0.600925213.Subproblem 1
y + -0.8333333335 = 0.600925213 Simplifying y + -0.8333333335 = 0.600925213 Reorder the terms: -0.8333333335 + y = 0.600925213 Solving -0.8333333335 + y = 0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = 0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = 0.600925213 + 0.8333333335 y = 0.600925213 + 0.8333333335 Combine like terms: 0.600925213 + 0.8333333335 = 1.4342585465 y = 1.4342585465 Simplifying y = 1.4342585465Subproblem 2
y + -0.8333333335 = -0.600925213 Simplifying y + -0.8333333335 = -0.600925213 Reorder the terms: -0.8333333335 + y = -0.600925213 Solving -0.8333333335 + y = -0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = -0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = -0.600925213 + 0.8333333335 y = -0.600925213 + 0.8333333335 Combine like terms: -0.600925213 + 0.8333333335 = 0.2324081205 y = 0.2324081205 Simplifying y = 0.2324081205Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.4342585465, 0.2324081205}Solution
y = {0, 1.4342585465, 0.2324081205}
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