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Simplifying 80y3 + -40y2 + -5y = 0 Reorder the terms: -5y + -40y2 + 80y3 = 0 Solving -5y + -40y2 + 80y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '5y'. 5y(-1 + -8y + 16y2) = 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 + -8y + 16y2)' equal to zero and attempt to solve: Simplifying -1 + -8y + 16y2 = 0 Solving -1 + -8y + 16y2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.0625 + -0.5y + y2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -0.5y + 0.0625 + y2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -0.5y + y2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -0.5y + y2 = 0 + 0.0625 -0.5y + y2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -0.5y + y2 = 0.0625 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 = 0.0625 + 0.0625 Reorder the terms: 0.0625 + -0.5y + y2 = 0.0625 + 0.0625 Combine like terms: 0.0625 + 0.0625 = 0.125 0.0625 + -0.5y + y2 = 0.125 Factor a perfect square on the left side: (y + -0.25)(y + -0.25) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (y + -0.25) equal to 0.353553391 and -0.353553391.Subproblem 1
y + -0.25 = 0.353553391 Simplifying y + -0.25 = 0.353553391 Reorder the terms: -0.25 + y = 0.353553391 Solving -0.25 + y = 0.353553391 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 = 0.353553391 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 0.353553391 + 0.25 y = 0.353553391 + 0.25 Combine like terms: 0.353553391 + 0.25 = 0.603553391 y = 0.603553391 Simplifying y = 0.603553391Subproblem 2
y + -0.25 = -0.353553391 Simplifying y + -0.25 = -0.353553391 Reorder the terms: -0.25 + y = -0.353553391 Solving -0.25 + y = -0.353553391 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 = -0.353553391 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -0.353553391 + 0.25 y = -0.353553391 + 0.25 Combine like terms: -0.353553391 + 0.25 = -0.103553391 y = -0.103553391 Simplifying y = -0.103553391Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.603553391, -0.103553391}Solution
y = {0, 0.603553391, -0.103553391}
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