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Simplifying (11w + 5)(6w + -18) = w Reorder the terms: (5 + 11w)(6w + -18) = w Reorder the terms: (5 + 11w)(-18 + 6w) = w Multiply (5 + 11w) * (-18 + 6w) (5(-18 + 6w) + 11w * (-18 + 6w)) = w ((-18 * 5 + 6w * 5) + 11w * (-18 + 6w)) = w ((-90 + 30w) + 11w * (-18 + 6w)) = w (-90 + 30w + (-18 * 11w + 6w * 11w)) = w (-90 + 30w + (-198w + 66w2)) = w Combine like terms: 30w + -198w = -168w (-90 + -168w + 66w2) = w Solving -90 + -168w + 66w2 = w Solving for variable 'w'. Reorder the terms: -90 + -168w + -1w + 66w2 = w + -1w Combine like terms: -168w + -1w = -169w -90 + -169w + 66w2 = w + -1w Combine like terms: w + -1w = 0 -90 + -169w + 66w2 = 0 Begin completing the square. Divide all terms by 66 the coefficient of the squared term: Divide each side by '66'. -1.363636364 + -2.560606061w + w2 = 0 Move the constant term to the right: Add '1.363636364' to each side of the equation. -1.363636364 + -2.560606061w + 1.363636364 + w2 = 0 + 1.363636364 Reorder the terms: -1.363636364 + 1.363636364 + -2.560606061w + w2 = 0 + 1.363636364 Combine like terms: -1.363636364 + 1.363636364 = 0.000000000 0.000000000 + -2.560606061w + w2 = 0 + 1.363636364 -2.560606061w + w2 = 0 + 1.363636364 Combine like terms: 0 + 1.363636364 = 1.363636364 -2.560606061w + w2 = 1.363636364 The w term is -2.560606061w. Take half its coefficient (-1.280303031). Square it (1.639175851) and add it to both sides. Add '1.639175851' to each side of the equation. -2.560606061w + 1.639175851 + w2 = 1.363636364 + 1.639175851 Reorder the terms: 1.639175851 + -2.560606061w + w2 = 1.363636364 + 1.639175851 Combine like terms: 1.363636364 + 1.639175851 = 3.002812215 1.639175851 + -2.560606061w + w2 = 3.002812215 Factor a perfect square on the left side: (w + -1.280303031)(w + -1.280303031) = 3.002812215 Calculate the square root of the right side: 1.732862434 Break this problem into two subproblems by setting (w + -1.280303031) equal to 1.732862434 and -1.732862434.Subproblem 1
w + -1.280303031 = 1.732862434 Simplifying w + -1.280303031 = 1.732862434 Reorder the terms: -1.280303031 + w = 1.732862434 Solving -1.280303031 + w = 1.732862434 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.280303031' to each side of the equation. -1.280303031 + 1.280303031 + w = 1.732862434 + 1.280303031 Combine like terms: -1.280303031 + 1.280303031 = 0.000000000 0.000000000 + w = 1.732862434 + 1.280303031 w = 1.732862434 + 1.280303031 Combine like terms: 1.732862434 + 1.280303031 = 3.013165465 w = 3.013165465 Simplifying w = 3.013165465Subproblem 2
w + -1.280303031 = -1.732862434 Simplifying w + -1.280303031 = -1.732862434 Reorder the terms: -1.280303031 + w = -1.732862434 Solving -1.280303031 + w = -1.732862434 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.280303031' to each side of the equation. -1.280303031 + 1.280303031 + w = -1.732862434 + 1.280303031 Combine like terms: -1.280303031 + 1.280303031 = 0.000000000 0.000000000 + w = -1.732862434 + 1.280303031 w = -1.732862434 + 1.280303031 Combine like terms: -1.732862434 + 1.280303031 = -0.452559403 w = -0.452559403 Simplifying w = -0.452559403Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.013165465, -0.452559403}
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