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Simplifying 4y4 + -53y3 + 144y2 = 0 Reorder the terms: 144y2 + -53y3 + 4y4 = 0 Solving 144y2 + -53y3 + 4y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y2'. y2(144 + -53y + 4y2) = 0Subproblem 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 '(144 + -53y + 4y2)' equal to zero and attempt to solve: Simplifying 144 + -53y + 4y2 = 0 Solving 144 + -53y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 36 + -13.25y + y2 = 0 Move the constant term to the right: Add '-36' to each side of the equation. 36 + -13.25y + -36 + y2 = 0 + -36 Reorder the terms: 36 + -36 + -13.25y + y2 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + -13.25y + y2 = 0 + -36 -13.25y + y2 = 0 + -36 Combine like terms: 0 + -36 = -36 -13.25y + y2 = -36 The y term is -13.25y. Take half its coefficient (-6.625). Square it (43.890625) and add it to both sides. Add '43.890625' to each side of the equation. -13.25y + 43.890625 + y2 = -36 + 43.890625 Reorder the terms: 43.890625 + -13.25y + y2 = -36 + 43.890625 Combine like terms: -36 + 43.890625 = 7.890625 43.890625 + -13.25y + y2 = 7.890625 Factor a perfect square on the left side: (y + -6.625)(y + -6.625) = 7.890625 Calculate the square root of the right side: 2.809025632 Break this problem into two subproblems by setting (y + -6.625) equal to 2.809025632 and -2.809025632.Subproblem 1
y + -6.625 = 2.809025632 Simplifying y + -6.625 = 2.809025632 Reorder the terms: -6.625 + y = 2.809025632 Solving -6.625 + y = 2.809025632 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.625' to each side of the equation. -6.625 + 6.625 + y = 2.809025632 + 6.625 Combine like terms: -6.625 + 6.625 = 0.000 0.000 + y = 2.809025632 + 6.625 y = 2.809025632 + 6.625 Combine like terms: 2.809025632 + 6.625 = 9.434025632 y = 9.434025632 Simplifying y = 9.434025632Subproblem 2
y + -6.625 = -2.809025632 Simplifying y + -6.625 = -2.809025632 Reorder the terms: -6.625 + y = -2.809025632 Solving -6.625 + y = -2.809025632 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.625' to each side of the equation. -6.625 + 6.625 + y = -2.809025632 + 6.625 Combine like terms: -6.625 + 6.625 = 0.000 0.000 + y = -2.809025632 + 6.625 y = -2.809025632 + 6.625 Combine like terms: -2.809025632 + 6.625 = 3.815974368 y = 3.815974368 Simplifying y = 3.815974368Solution
The solution to the problem is based on the solutions from the subproblems. y = {9.434025632, 3.815974368}Solution
y = {0, 9.434025632, 3.815974368}
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