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Simplifying 6w2 + 4w + -2500 = 0 Reorder the terms: -2500 + 4w + 6w2 = 0 Solving -2500 + 4w + 6w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1250 + 2w + 3w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1250 + 2w + 3w2)' equal to zero and attempt to solve: Simplifying -1250 + 2w + 3w2 = 0 Solving -1250 + 2w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -416.6666667 + 0.6666666667w + w2 = 0 Move the constant term to the right: Add '416.6666667' to each side of the equation. -416.6666667 + 0.6666666667w + 416.6666667 + w2 = 0 + 416.6666667 Reorder the terms: -416.6666667 + 416.6666667 + 0.6666666667w + w2 = 0 + 416.6666667 Combine like terms: -416.6666667 + 416.6666667 = 0.0000000 0.0000000 + 0.6666666667w + w2 = 0 + 416.6666667 0.6666666667w + w2 = 0 + 416.6666667 Combine like terms: 0 + 416.6666667 = 416.6666667 0.6666666667w + w2 = 416.6666667 The w term is 0.6666666667w. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667w + 0.1111111112 + w2 = 416.6666667 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667w + w2 = 416.6666667 + 0.1111111112 Combine like terms: 416.6666667 + 0.1111111112 = 416.7777778112 0.1111111112 + 0.6666666667w + w2 = 416.7777778112 Factor a perfect square on the left side: (w + 0.3333333334)(w + 0.3333333334) = 416.7777778112 Calculate the square root of the right side: 20.415135998 Break this problem into two subproblems by setting (w + 0.3333333334) equal to 20.415135998 and -20.415135998.Subproblem 1
w + 0.3333333334 = 20.415135998 Simplifying w + 0.3333333334 = 20.415135998 Reorder the terms: 0.3333333334 + w = 20.415135998 Solving 0.3333333334 + w = 20.415135998 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + w = 20.415135998 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + w = 20.415135998 + -0.3333333334 w = 20.415135998 + -0.3333333334 Combine like terms: 20.415135998 + -0.3333333334 = 20.0818026646 w = 20.0818026646 Simplifying w = 20.0818026646Subproblem 2
w + 0.3333333334 = -20.415135998 Simplifying w + 0.3333333334 = -20.415135998 Reorder the terms: 0.3333333334 + w = -20.415135998 Solving 0.3333333334 + w = -20.415135998 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + w = -20.415135998 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + w = -20.415135998 + -0.3333333334 w = -20.415135998 + -0.3333333334 Combine like terms: -20.415135998 + -0.3333333334 = -20.7484693314 w = -20.7484693314 Simplifying w = -20.7484693314Solution
The solution to the problem is based on the solutions from the subproblems. w = {20.0818026646, -20.7484693314}Solution
w = {20.0818026646, -20.7484693314}
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