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Simplifying 9w2 + -33w + 22 = 0 Reorder the terms: 22 + -33w + 9w2 = 0 Solving 22 + -33w + 9w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.444444444 + -3.666666667w + w2 = 0 Move the constant term to the right: Add '-2.444444444' to each side of the equation. 2.444444444 + -3.666666667w + -2.444444444 + w2 = 0 + -2.444444444 Reorder the terms: 2.444444444 + -2.444444444 + -3.666666667w + w2 = 0 + -2.444444444 Combine like terms: 2.444444444 + -2.444444444 = 0.000000000 0.000000000 + -3.666666667w + w2 = 0 + -2.444444444 -3.666666667w + w2 = 0 + -2.444444444 Combine like terms: 0 + -2.444444444 = -2.444444444 -3.666666667w + w2 = -2.444444444 The w term is -3.666666667w. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667w + 3.361111114 + w2 = -2.444444444 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667w + w2 = -2.444444444 + 3.361111114 Combine like terms: -2.444444444 + 3.361111114 = 0.91666667 3.361111114 + -3.666666667w + w2 = 0.91666667 Factor a perfect square on the left side: (w + -1.833333334)(w + -1.833333334) = 0.91666667 Calculate the square root of the right side: 0.957427109 Break this problem into two subproblems by setting (w + -1.833333334) equal to 0.957427109 and -0.957427109.Subproblem 1
w + -1.833333334 = 0.957427109 Simplifying w + -1.833333334 = 0.957427109 Reorder the terms: -1.833333334 + w = 0.957427109 Solving -1.833333334 + w = 0.957427109 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + w = 0.957427109 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + w = 0.957427109 + 1.833333334 w = 0.957427109 + 1.833333334 Combine like terms: 0.957427109 + 1.833333334 = 2.790760443 w = 2.790760443 Simplifying w = 2.790760443Subproblem 2
w + -1.833333334 = -0.957427109 Simplifying w + -1.833333334 = -0.957427109 Reorder the terms: -1.833333334 + w = -0.957427109 Solving -1.833333334 + w = -0.957427109 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + w = -0.957427109 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + w = -0.957427109 + 1.833333334 w = -0.957427109 + 1.833333334 Combine like terms: -0.957427109 + 1.833333334 = 0.875906225 w = 0.875906225 Simplifying w = 0.875906225Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.790760443, 0.875906225}
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