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Simplifying 2w2 + -19w + 31 = 49 + -4w2 + -4 Reorder the terms: 31 + -19w + 2w2 = 49 + -4w2 + -4 Reorder the terms: 31 + -19w + 2w2 = 49 + -4 + -4w2 Combine like terms: 49 + -4 = 45 31 + -19w + 2w2 = 45 + -4w2 Solving 31 + -19w + 2w2 = 45 + -4w2 Solving for variable 'w'. Reorder the terms: 31 + -45 + -19w + 2w2 + 4w2 = 45 + -4w2 + -45 + 4w2 Combine like terms: 31 + -45 = -14 -14 + -19w + 2w2 + 4w2 = 45 + -4w2 + -45 + 4w2 Combine like terms: 2w2 + 4w2 = 6w2 -14 + -19w + 6w2 = 45 + -4w2 + -45 + 4w2 Reorder the terms: -14 + -19w + 6w2 = 45 + -45 + -4w2 + 4w2 Combine like terms: 45 + -45 = 0 -14 + -19w + 6w2 = 0 + -4w2 + 4w2 -14 + -19w + 6w2 = -4w2 + 4w2 Combine like terms: -4w2 + 4w2 = 0 -14 + -19w + 6w2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.333333333 + -3.166666667w + w2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -3.166666667w + 2.333333333 + w2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -3.166666667w + w2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -3.166666667w + w2 = 0 + 2.333333333 -3.166666667w + w2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -3.166666667w + w2 = 2.333333333 The w term is -3.166666667w. Take half its coefficient (-1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. -3.166666667w + 2.506944447 + w2 = 2.333333333 + 2.506944447 Reorder the terms: 2.506944447 + -3.166666667w + w2 = 2.333333333 + 2.506944447 Combine like terms: 2.333333333 + 2.506944447 = 4.84027778 2.506944447 + -3.166666667w + w2 = 4.84027778 Factor a perfect square on the left side: (w + -1.583333334)(w + -1.583333334) = 4.84027778 Calculate the square root of the right side: 2.200063131 Break this problem into two subproblems by setting (w + -1.583333334) equal to 2.200063131 and -2.200063131.Subproblem 1
w + -1.583333334 = 2.200063131 Simplifying w + -1.583333334 = 2.200063131 Reorder the terms: -1.583333334 + w = 2.200063131 Solving -1.583333334 + w = 2.200063131 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + w = 2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + w = 2.200063131 + 1.583333334 w = 2.200063131 + 1.583333334 Combine like terms: 2.200063131 + 1.583333334 = 3.783396465 w = 3.783396465 Simplifying w = 3.783396465Subproblem 2
w + -1.583333334 = -2.200063131 Simplifying w + -1.583333334 = -2.200063131 Reorder the terms: -1.583333334 + w = -2.200063131 Solving -1.583333334 + w = -2.200063131 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + w = -2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + w = -2.200063131 + 1.583333334 w = -2.200063131 + 1.583333334 Combine like terms: -2.200063131 + 1.583333334 = -0.616729797 w = -0.616729797 Simplifying w = -0.616729797Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.783396465, -0.616729797}
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