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Simplifying (4W2 + -6W + 2) + (-7W2 + 3W + 1) = 0 Reorder the terms: (2 + -6W + 4W2) + (-7W2 + 3W + 1) = 0 Remove parenthesis around (2 + -6W + 4W2) 2 + -6W + 4W2 + (-7W2 + 3W + 1) = 0 Reorder the terms: 2 + -6W + 4W2 + (1 + 3W + -7W2) = 0 Remove parenthesis around (1 + 3W + -7W2) 2 + -6W + 4W2 + 1 + 3W + -7W2 = 0 Reorder the terms: 2 + 1 + -6W + 3W + 4W2 + -7W2 = 0 Combine like terms: 2 + 1 = 3 3 + -6W + 3W + 4W2 + -7W2 = 0 Combine like terms: -6W + 3W = -3W 3 + -3W + 4W2 + -7W2 = 0 Combine like terms: 4W2 + -7W2 = -3W2 3 + -3W + -3W2 = 0 Solving 3 + -3W + -3W2 = 0 Solving for variable 'W'. Factor out the Greatest Common Factor (GCF), '3'. 3(1 + -1W + -1W2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + -1W + -1W2)' equal to zero and attempt to solve: Simplifying 1 + -1W + -1W2 = 0 Solving 1 + -1W + -1W2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + W + W2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + W + 1 + W2 = 0 + 1 Reorder the terms: -1 + 1 + W + W2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + W + W2 = 0 + 1 W + W2 = 0 + 1 Combine like terms: 0 + 1 = 1 W + W2 = 1 The W term is W. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + W2 = 1 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + W2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 1.25 + W2 = 1.25 Factor a perfect square on the left side: (W + 0.5)(W + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (W + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
W + 0.5 = 1.118033989 Simplifying W + 0.5 = 1.118033989 Reorder the terms: 0.5 + W = 1.118033989 Solving 0.5 + W = 1.118033989 Solving for variable 'W'. Move all terms containing W to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + W = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + W = 1.118033989 + -0.5 W = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 W = 0.618033989 Simplifying W = 0.618033989Subproblem 2
W + 0.5 = -1.118033989 Simplifying W + 0.5 = -1.118033989 Reorder the terms: 0.5 + W = -1.118033989 Solving 0.5 + W = -1.118033989 Solving for variable 'W'. Move all terms containing W to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + W = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + W = -1.118033989 + -0.5 W = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 W = -1.618033989 Simplifying W = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. W = {0.618033989, -1.618033989}Solution
W = {0.618033989, -1.618033989}
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