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Simplifying y5 + -16y4 + 52y3 = 0 Reorder the terms: 52y3 + -16y4 + y5 = 0 Solving 52y3 + -16y4 + y5 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y3'. y3(52 + -16y + y2) = 0Subproblem 1
Set the factor 'y3' equal to zero and attempt to solve: Simplifying y3 = 0 Solving y3 = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y3 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(52 + -16y + y2)' equal to zero and attempt to solve: Simplifying 52 + -16y + y2 = 0 Solving 52 + -16y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-52' to each side of the equation. 52 + -16y + -52 + y2 = 0 + -52 Reorder the terms: 52 + -52 + -16y + y2 = 0 + -52 Combine like terms: 52 + -52 = 0 0 + -16y + y2 = 0 + -52 -16y + y2 = 0 + -52 Combine like terms: 0 + -52 = -52 -16y + y2 = -52 The y term is -16y. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16y + 64 + y2 = -52 + 64 Reorder the terms: 64 + -16y + y2 = -52 + 64 Combine like terms: -52 + 64 = 12 64 + -16y + y2 = 12 Factor a perfect square on the left side: (y + -8)(y + -8) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (y + -8) equal to 3.464101615 and -3.464101615.Subproblem 1
y + -8 = 3.464101615 Simplifying y + -8 = 3.464101615 Reorder the terms: -8 + y = 3.464101615 Solving -8 + y = 3.464101615 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = 3.464101615 + 8 Combine like terms: -8 + 8 = 0 0 + y = 3.464101615 + 8 y = 3.464101615 + 8 Combine like terms: 3.464101615 + 8 = 11.464101615 y = 11.464101615 Simplifying y = 11.464101615Subproblem 2
y + -8 = -3.464101615 Simplifying y + -8 = -3.464101615 Reorder the terms: -8 + y = -3.464101615 Solving -8 + y = -3.464101615 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = -3.464101615 + 8 Combine like terms: -8 + 8 = 0 0 + y = -3.464101615 + 8 y = -3.464101615 + 8 Combine like terms: -3.464101615 + 8 = 4.535898385 y = 4.535898385 Simplifying y = 4.535898385Solution
The solution to the problem is based on the solutions from the subproblems. y = {11.464101615, 4.535898385}Solution
y = {11.464101615, 4.535898385}
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