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Simplifying 24y4 + -40y3 + -32y2 = 0 Reorder the terms: -32y2 + -40y3 + 24y4 = 0 Solving -32y2 + -40y3 + 24y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '8y2'. 8y2(-4 + -5y + 3y2) = 0 Ignore the factor 8.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 '(-4 + -5y + 3y2)' equal to zero and attempt to solve: Simplifying -4 + -5y + 3y2 = 0 Solving -4 + -5y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.333333333 + -1.666666667y + y2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -1.666666667y + 1.333333333 + y2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -1.666666667y + y2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -1.666666667y + y2 = 0 + 1.333333333 -1.666666667y + y2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -1.666666667y + y2 = 1.333333333 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 = 1.333333333 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667y + y2 = 1.333333333 + 0.6944444447 Combine like terms: 1.333333333 + 0.6944444447 = 2.0277777777 0.6944444447 + -1.666666667y + y2 = 2.0277777777 Factor a perfect square on the left side: (y + -0.8333333335)(y + -0.8333333335) = 2.0277777777 Calculate the square root of the right side: 1.424000624 Break this problem into two subproblems by setting (y + -0.8333333335) equal to 1.424000624 and -1.424000624.Subproblem 1
y + -0.8333333335 = 1.424000624 Simplifying y + -0.8333333335 = 1.424000624 Reorder the terms: -0.8333333335 + y = 1.424000624 Solving -0.8333333335 + y = 1.424000624 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 = 1.424000624 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = 1.424000624 + 0.8333333335 y = 1.424000624 + 0.8333333335 Combine like terms: 1.424000624 + 0.8333333335 = 2.2573339575 y = 2.2573339575 Simplifying y = 2.2573339575Subproblem 2
y + -0.8333333335 = -1.424000624 Simplifying y + -0.8333333335 = -1.424000624 Reorder the terms: -0.8333333335 + y = -1.424000624 Solving -0.8333333335 + y = -1.424000624 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 = -1.424000624 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = -1.424000624 + 0.8333333335 y = -1.424000624 + 0.8333333335 Combine like terms: -1.424000624 + 0.8333333335 = -0.5906672905 y = -0.5906672905 Simplifying y = -0.5906672905Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.2573339575, -0.5906672905}Solution
y = {0, 2.2573339575, -0.5906672905}
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