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Simplifying 6w4 + w3 + -3w2 = 0 Reorder the terms: -3w2 + w3 + 6w4 = 0 Solving -3w2 + w3 + 6w4 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), 'w2'. w2(-3 + w + 6w2) = 0Subproblem 1
Set the factor 'w2' equal to zero and attempt to solve: Simplifying w2 = 0 Solving w2 = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w2 = 0 Take the square root of each side: w = {0}Subproblem 2
Set the factor '(-3 + w + 6w2)' equal to zero and attempt to solve: Simplifying -3 + w + 6w2 = 0 Solving -3 + w + 6w2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.1666666667w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.1666666667w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.1666666667w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.1666666667w + w2 = 0 + 0.5 0.1666666667w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.1666666667w + w2 = 0.5 The w term is w. 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. 0.1666666667w + 0.25 + w2 = 0.5 + 0.25 Reorder the terms: 0.25 + 0.1666666667w + w2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + 0.1666666667w + w2 = 0.75 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (w + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
w + 0.5 = 0.866025404 Simplifying w + 0.5 = 0.866025404 Reorder the terms: 0.5 + w = 0.866025404 Solving 0.5 + w = 0.866025404 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.866025404 + -0.5 w = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 w = 0.366025404 Simplifying w = 0.366025404Subproblem 2
w + 0.5 = -0.866025404 Simplifying w + 0.5 = -0.866025404 Reorder the terms: 0.5 + w = -0.866025404 Solving 0.5 + w = -0.866025404 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -0.866025404 + -0.5 w = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 w = -1.366025404 Simplifying w = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.366025404, -1.366025404}Solution
w = {0, 0.366025404, -1.366025404}
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