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Simplifying 18w3 + -39w2 + -20w = 0 Reorder the terms: -20w + -39w2 + 18w3 = 0 Solving -20w + -39w2 + 18w3 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), 'w'. w(-20 + -39w + 18w2) = 0Subproblem 1
Set the factor 'w' equal to zero and attempt to solve: Simplifying w = 0 Solving w = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w = 0Subproblem 2
Set the factor '(-20 + -39w + 18w2)' equal to zero and attempt to solve: Simplifying -20 + -39w + 18w2 = 0 Solving -20 + -39w + 18w2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.111111111 + -2.166666667w + w2 = 0 Move the constant term to the right: Add '1.111111111' to each side of the equation. -1.111111111 + -2.166666667w + 1.111111111 + w2 = 0 + 1.111111111 Reorder the terms: -1.111111111 + 1.111111111 + -2.166666667w + w2 = 0 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + -2.166666667w + w2 = 0 + 1.111111111 -2.166666667w + w2 = 0 + 1.111111111 Combine like terms: 0 + 1.111111111 = 1.111111111 -2.166666667w + w2 = 1.111111111 The w term is -2.166666667w. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667w + 1.173611113 + w2 = 1.111111111 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667w + w2 = 1.111111111 + 1.173611113 Combine like terms: 1.111111111 + 1.173611113 = 2.284722224 1.173611113 + -2.166666667w + w2 = 2.284722224 Factor a perfect square on the left side: (w + -1.083333334)(w + -1.083333334) = 2.284722224 Calculate the square root of the right side: 1.511529763 Break this problem into two subproblems by setting (w + -1.083333334) equal to 1.511529763 and -1.511529763.Subproblem 1
w + -1.083333334 = 1.511529763 Simplifying w + -1.083333334 = 1.511529763 Reorder the terms: -1.083333334 + w = 1.511529763 Solving -1.083333334 + w = 1.511529763 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + w = 1.511529763 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + w = 1.511529763 + 1.083333334 w = 1.511529763 + 1.083333334 Combine like terms: 1.511529763 + 1.083333334 = 2.594863097 w = 2.594863097 Simplifying w = 2.594863097Subproblem 2
w + -1.083333334 = -1.511529763 Simplifying w + -1.083333334 = -1.511529763 Reorder the terms: -1.083333334 + w = -1.511529763 Solving -1.083333334 + w = -1.511529763 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + w = -1.511529763 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + w = -1.511529763 + 1.083333334 w = -1.511529763 + 1.083333334 Combine like terms: -1.511529763 + 1.083333334 = -0.428196429 w = -0.428196429 Simplifying w = -0.428196429Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.594863097, -0.428196429}Solution
w = {0, 2.594863097, -0.428196429}
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