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Simplifying -8w2 + w3 + 6w4 = 0 Solving -8w2 + w3 + 6w4 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), 'w2'. w2(-8 + 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 '(-8 + w + 6w2)' equal to zero and attempt to solve: Simplifying -8 + w + 6w2 = 0 Solving -8 + w + 6w2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 0.1666666667w + w2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 0.1666666667w + 1.333333333 + w2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 0.1666666667w + w2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 0.1666666667w + w2 = 0 + 1.333333333 0.1666666667w + w2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 0.1666666667w + w2 = 1.333333333 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 = 1.333333333 + 0.25 Reorder the terms: 0.25 + 0.1666666667w + w2 = 1.333333333 + 0.25 Combine like terms: 1.333333333 + 0.25 = 1.583333333 0.25 + 0.1666666667w + w2 = 1.583333333 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (w + 0.5) equal to 1.258305739 and -1.258305739.Subproblem 1
w + 0.5 = 1.258305739 Simplifying w + 0.5 = 1.258305739 Reorder the terms: 0.5 + w = 1.258305739 Solving 0.5 + w = 1.258305739 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 = 1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 1.258305739 + -0.5 w = 1.258305739 + -0.5 Combine like terms: 1.258305739 + -0.5 = 0.758305739 w = 0.758305739 Simplifying w = 0.758305739Subproblem 2
w + 0.5 = -1.258305739 Simplifying w + 0.5 = -1.258305739 Reorder the terms: 0.5 + w = -1.258305739 Solving 0.5 + w = -1.258305739 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 = -1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -1.258305739 + -0.5 w = -1.258305739 + -0.5 Combine like terms: -1.258305739 + -0.5 = -1.758305739 w = -1.758305739 Simplifying w = -1.758305739Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.758305739, -1.758305739}Solution
w = {0, 0.758305739, -1.758305739}
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