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Simplifying 5y3 + y2 + -16y = 0 Reorder the terms: -16y + y2 + 5y3 = 0 Solving -16y + y2 + 5y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-16 + y + 5y2) = 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 '(-16 + y + 5y2)' equal to zero and attempt to solve: Simplifying -16 + y + 5y2 = 0 Solving -16 + y + 5y2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -3.2 + 0.2y + y2 = 0 Move the constant term to the right: Add '3.2' to each side of the equation. -3.2 + 0.2y + 3.2 + y2 = 0 + 3.2 Reorder the terms: -3.2 + 3.2 + 0.2y + y2 = 0 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + 0.2y + y2 = 0 + 3.2 0.2y + y2 = 0 + 3.2 Combine like terms: 0 + 3.2 = 3.2 0.2y + y2 = 3.2 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. 0.2y + 0.25 + y2 = 3.2 + 0.25 Reorder the terms: 0.25 + 0.2y + y2 = 3.2 + 0.25 Combine like terms: 3.2 + 0.25 = 3.45 0.25 + 0.2y + y2 = 3.45 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 3.45 Calculate the square root of the right side: 1.857417562 Break this problem into two subproblems by setting (y + 0.5) equal to 1.857417562 and -1.857417562.Subproblem 1
y + 0.5 = 1.857417562 Simplifying y + 0.5 = 1.857417562 Reorder the terms: 0.5 + y = 1.857417562 Solving 0.5 + y = 1.857417562 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 = 1.857417562 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.857417562 + -0.5 y = 1.857417562 + -0.5 Combine like terms: 1.857417562 + -0.5 = 1.357417562 y = 1.357417562 Simplifying y = 1.357417562Subproblem 2
y + 0.5 = -1.857417562 Simplifying y + 0.5 = -1.857417562 Reorder the terms: 0.5 + y = -1.857417562 Solving 0.5 + y = -1.857417562 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 = -1.857417562 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.857417562 + -0.5 y = -1.857417562 + -0.5 Combine like terms: -1.857417562 + -0.5 = -2.357417562 y = -2.357417562 Simplifying y = -2.357417562Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.357417562, -2.357417562}Solution
y = {0, 1.357417562, -2.357417562}
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