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Simplifying w(3w + -7) = 120 Reorder the terms: w(-7 + 3w) = 120 (-7 * w + 3w * w) = 120 (-7w + 3w2) = 120 Solving -7w + 3w2 = 120 Solving for variable 'w'. Reorder the terms: -120 + -7w + 3w2 = 120 + -120 Combine like terms: 120 + -120 = 0 -120 + -7w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -40 + -2.333333333w + w2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + -2.333333333w + 40 + w2 = 0 + 40 Reorder the terms: -40 + 40 + -2.333333333w + w2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -2.333333333w + w2 = 0 + 40 -2.333333333w + w2 = 0 + 40 Combine like terms: 0 + 40 = 40 -2.333333333w + w2 = 40 The w term is -2.333333333w. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333w + 1.361111112 + w2 = 40 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333w + w2 = 40 + 1.361111112 Combine like terms: 40 + 1.361111112 = 41.361111112 1.361111112 + -2.333333333w + w2 = 41.361111112 Factor a perfect square on the left side: (w + -1.166666667)(w + -1.166666667) = 41.361111112 Calculate the square root of the right side: 6.431260461 Break this problem into two subproblems by setting (w + -1.166666667) equal to 6.431260461 and -6.431260461.Subproblem 1
w + -1.166666667 = 6.431260461 Simplifying w + -1.166666667 = 6.431260461 Reorder the terms: -1.166666667 + w = 6.431260461 Solving -1.166666667 + w = 6.431260461 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + w = 6.431260461 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + w = 6.431260461 + 1.166666667 w = 6.431260461 + 1.166666667 Combine like terms: 6.431260461 + 1.166666667 = 7.597927128 w = 7.597927128 Simplifying w = 7.597927128Subproblem 2
w + -1.166666667 = -6.431260461 Simplifying w + -1.166666667 = -6.431260461 Reorder the terms: -1.166666667 + w = -6.431260461 Solving -1.166666667 + w = -6.431260461 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + w = -6.431260461 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + w = -6.431260461 + 1.166666667 w = -6.431260461 + 1.166666667 Combine like terms: -6.431260461 + 1.166666667 = -5.264593794 w = -5.264593794 Simplifying w = -5.264593794Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.597927128, -5.264593794}
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