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Simplifying y3 + y2 + -4y = 0 Reorder the terms: -4y + y2 + y3 = 0 Solving -4y + y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-4 + y + y2) = 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 '(-4 + y + y2)' equal to zero and attempt to solve: Simplifying -4 + y + y2 = 0 Solving -4 + y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + y + 4 + y2 = 0 + 4 Reorder the terms: -4 + 4 + y + y2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + y + y2 = 0 + 4 y + y2 = 0 + 4 Combine like terms: 0 + 4 = 4 y + y2 = 4 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. y + 0.25 + y2 = 4 + 0.25 Reorder the terms: 0.25 + y + y2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + y + y2 = 4.25 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (y + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
y + 0.5 = 2.061552813 Simplifying y + 0.5 = 2.061552813 Reorder the terms: 0.5 + y = 2.061552813 Solving 0.5 + y = 2.061552813 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 2.061552813 + -0.5 y = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 y = 1.561552813 Simplifying y = 1.561552813Subproblem 2
y + 0.5 = -2.061552813 Simplifying y + 0.5 = -2.061552813 Reorder the terms: 0.5 + y = -2.061552813 Solving 0.5 + y = -2.061552813 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -2.061552813 + -0.5 y = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 y = -2.561552813 Simplifying y = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.561552813, -2.561552813}Solution
y = {0, 1.561552813, -2.561552813}
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