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Simplifying 112 = 2w2 + 6w + -20 Reorder the terms: 112 = -20 + 6w + 2w2 Solving 112 = -20 + 6w + 2w2 Solving for variable 'w'. Combine like terms: 112 + 20 = 132 132 + -6w + -2w2 = -20 + 6w + 2w2 + 20 + -6w + -2w2 Reorder the terms: 132 + -6w + -2w2 = -20 + 20 + 6w + -6w + 2w2 + -2w2 Combine like terms: -20 + 20 = 0 132 + -6w + -2w2 = 0 + 6w + -6w + 2w2 + -2w2 132 + -6w + -2w2 = 6w + -6w + 2w2 + -2w2 Combine like terms: 6w + -6w = 0 132 + -6w + -2w2 = 0 + 2w2 + -2w2 132 + -6w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 132 + -6w + -2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(66 + -3w + -1w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(66 + -3w + -1w2)' equal to zero and attempt to solve: Simplifying 66 + -3w + -1w2 = 0 Solving 66 + -3w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -66 + 3w + w2 = 0 Move the constant term to the right: Add '66' to each side of the equation. -66 + 3w + 66 + w2 = 0 + 66 Reorder the terms: -66 + 66 + 3w + w2 = 0 + 66 Combine like terms: -66 + 66 = 0 0 + 3w + w2 = 0 + 66 3w + w2 = 0 + 66 Combine like terms: 0 + 66 = 66 3w + w2 = 66 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 66 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 66 + 2.25 Combine like terms: 66 + 2.25 = 68.25 2.25 + 3w + w2 = 68.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 68.25 Calculate the square root of the right side: 8.261355821 Break this problem into two subproblems by setting (w + 1.5) equal to 8.261355821 and -8.261355821.Subproblem 1
w + 1.5 = 8.261355821 Simplifying w + 1.5 = 8.261355821 Reorder the terms: 1.5 + w = 8.261355821 Solving 1.5 + w = 8.261355821 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 8.261355821 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 8.261355821 + -1.5 w = 8.261355821 + -1.5 Combine like terms: 8.261355821 + -1.5 = 6.761355821 w = 6.761355821 Simplifying w = 6.761355821Subproblem 2
w + 1.5 = -8.261355821 Simplifying w + 1.5 = -8.261355821 Reorder the terms: 1.5 + w = -8.261355821 Solving 1.5 + w = -8.261355821 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -8.261355821 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -8.261355821 + -1.5 w = -8.261355821 + -1.5 Combine like terms: -8.261355821 + -1.5 = -9.761355821 w = -9.761355821 Simplifying w = -9.761355821Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.761355821, -9.761355821}Solution
w = {6.761355821, -9.761355821}
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