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Simplifying 4y2 + 4y3 + 24y4 = 0 Solving 4y2 + 4y3 + 24y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '4y2'. 4y2(1 + y + 6y2) = 0 Ignore the factor 4.Subproblem 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 '(1 + y + 6y2)' equal to zero and attempt to solve: Simplifying 1 + y + 6y2 = 0 Solving 1 + y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 0.1666666667y + y2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 0.1666666667y + -0.1666666667 + y2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 0.1666666667y + y2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 0.1666666667y + y2 = 0 + -0.1666666667 0.1666666667y + y2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 0.1666666667y + y2 = -0.1666666667 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. 0.1666666667y + 0.25 + y2 = -0.1666666667 + 0.25 Reorder the terms: 0.25 + 0.1666666667y + y2 = -0.1666666667 + 0.25 Combine like terms: -0.1666666667 + 0.25 = 0.0833333333 0.25 + 0.1666666667y + y2 = 0.0833333333 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 0.0833333333 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (y + 0.5) equal to 0.288675135 and -0.288675135.Subproblem 1
y + 0.5 = 0.288675135 Simplifying y + 0.5 = 0.288675135 Reorder the terms: 0.5 + y = 0.288675135 Solving 0.5 + y = 0.288675135 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 = 0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 0.288675135 + -0.5 y = 0.288675135 + -0.5 Combine like terms: 0.288675135 + -0.5 = -0.211324865 y = -0.211324865 Simplifying y = -0.211324865Subproblem 2
y + 0.5 = -0.288675135 Simplifying y + 0.5 = -0.288675135 Reorder the terms: 0.5 + y = -0.288675135 Solving 0.5 + y = -0.288675135 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 = -0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -0.288675135 + -0.5 y = -0.288675135 + -0.5 Combine like terms: -0.288675135 + -0.5 = -0.788675135 y = -0.788675135 Simplifying y = -0.788675135Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.211324865, -0.788675135}Solution
y = {0, -0.211324865, -0.788675135}
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