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Simplifying 25y3 = y + 75y2 Solving 25y3 = y + 75y2 Solving for variable 'y'. Reorder the terms: -1y + -75y2 + 25y3 = y + 75y2 + -1y + -75y2 Reorder the terms: -1y + -75y2 + 25y3 = y + -1y + 75y2 + -75y2 Combine like terms: y + -1y = 0 -1y + -75y2 + 25y3 = 0 + 75y2 + -75y2 -1y + -75y2 + 25y3 = 75y2 + -75y2 Combine like terms: 75y2 + -75y2 = 0 -1y + -75y2 + 25y3 = 0 Factor out the Greatest Common Factor (GCF), 'y'. y(-1 + -75y + 25y2) = 0Subproblem 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 + -75y + 25y2)' equal to zero and attempt to solve: Simplifying -1 + -75y + 25y2 = 0 Solving -1 + -75y + 25y2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.04 + -3y + y2 = 0 Move the constant term to the right: Add '0.04' to each side of the equation. -0.04 + -3y + 0.04 + y2 = 0 + 0.04 Reorder the terms: -0.04 + 0.04 + -3y + y2 = 0 + 0.04 Combine like terms: -0.04 + 0.04 = 0.00 0.00 + -3y + y2 = 0 + 0.04 -3y + y2 = 0 + 0.04 Combine like terms: 0 + 0.04 = 0.04 -3y + y2 = 0.04 The y term is -3y. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3y + 2.25 + y2 = 0.04 + 2.25 Reorder the terms: 2.25 + -3y + y2 = 0.04 + 2.25 Combine like terms: 0.04 + 2.25 = 2.29 2.25 + -3y + y2 = 2.29 Factor a perfect square on the left side: (y + -1.5)(y + -1.5) = 2.29 Calculate the square root of the right side: 1.513274595 Break this problem into two subproblems by setting (y + -1.5) equal to 1.513274595 and -1.513274595.Subproblem 1
y + -1.5 = 1.513274595 Simplifying y + -1.5 = 1.513274595 Reorder the terms: -1.5 + y = 1.513274595 Solving -1.5 + y = 1.513274595 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = 1.513274595 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 1.513274595 + 1.5 y = 1.513274595 + 1.5 Combine like terms: 1.513274595 + 1.5 = 3.013274595 y = 3.013274595 Simplifying y = 3.013274595Subproblem 2
y + -1.5 = -1.513274595 Simplifying y + -1.5 = -1.513274595 Reorder the terms: -1.5 + y = -1.513274595 Solving -1.5 + y = -1.513274595 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = -1.513274595 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -1.513274595 + 1.5 y = -1.513274595 + 1.5 Combine like terms: -1.513274595 + 1.5 = -0.013274595 y = -0.013274595 Simplifying y = -0.013274595Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.013274595, -0.013274595}Solution
y = {0, 3.013274595, -0.013274595}
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