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Simplifying 5y3 + 10y2 + -25y = 0 Reorder the terms: -25y + 10y2 + 5y3 = 0 Solving -25y + 10y2 + 5y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '5y'. 5y(-5 + 2y + y2) = 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 '(-5 + 2y + y2)' equal to zero and attempt to solve: Simplifying -5 + 2y + y2 = 0 Solving -5 + 2y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 2y + 5 + y2 = 0 + 5 Reorder the terms: -5 + 5 + 2y + y2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 2y + y2 = 0 + 5 2y + y2 = 0 + 5 Combine like terms: 0 + 5 = 5 2y + y2 = 5 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 5 + 1 Reorder the terms: 1 + 2y + y2 = 5 + 1 Combine like terms: 5 + 1 = 6 1 + 2y + y2 = 6 Factor a perfect square on the left side: (y + 1)(y + 1) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (y + 1) equal to 2.449489743 and -2.449489743.Subproblem 1
y + 1 = 2.449489743 Simplifying y + 1 = 2.449489743 Reorder the terms: 1 + y = 2.449489743 Solving 1 + y = 2.449489743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 2.449489743 + -1 Combine like terms: 1 + -1 = 0 0 + y = 2.449489743 + -1 y = 2.449489743 + -1 Combine like terms: 2.449489743 + -1 = 1.449489743 y = 1.449489743 Simplifying y = 1.449489743Subproblem 2
y + 1 = -2.449489743 Simplifying y + 1 = -2.449489743 Reorder the terms: 1 + y = -2.449489743 Solving 1 + y = -2.449489743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -2.449489743 + -1 Combine like terms: 1 + -1 = 0 0 + y = -2.449489743 + -1 y = -2.449489743 + -1 Combine like terms: -2.449489743 + -1 = -3.449489743 y = -3.449489743 Simplifying y = -3.449489743Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.449489743, -3.449489743}Solution
y = {0, 1.449489743, -3.449489743}
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