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Simplifying (w2 + -4w + 1) + (5 + -5w2 + -3w) = 0 Reorder the terms: (1 + -4w + w2) + (5 + -5w2 + -3w) = 0 Remove parenthesis around (1 + -4w + w2) 1 + -4w + w2 + (5 + -5w2 + -3w) = 0 Reorder the terms: 1 + -4w + w2 + (5 + -3w + -5w2) = 0 Remove parenthesis around (5 + -3w + -5w2) 1 + -4w + w2 + 5 + -3w + -5w2 = 0 Reorder the terms: 1 + 5 + -4w + -3w + w2 + -5w2 = 0 Combine like terms: 1 + 5 = 6 6 + -4w + -3w + w2 + -5w2 = 0 Combine like terms: -4w + -3w = -7w 6 + -7w + w2 + -5w2 = 0 Combine like terms: w2 + -5w2 = -4w2 6 + -7w + -4w2 = 0 Solving 6 + -7w + -4w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.5 + 1.75w + w2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.75w + 1.5 + w2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.75w + w2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.75w + w2 = 0 + 1.5 1.75w + w2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.75w + w2 = 1.5 The w term is 1.75w. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75w + 0.765625 + w2 = 1.5 + 0.765625 Reorder the terms: 0.765625 + 1.75w + w2 = 1.5 + 0.765625 Combine like terms: 1.5 + 0.765625 = 2.265625 0.765625 + 1.75w + w2 = 2.265625 Factor a perfect square on the left side: (w + 0.875)(w + 0.875) = 2.265625 Calculate the square root of the right side: 1.505199322 Break this problem into two subproblems by setting (w + 0.875) equal to 1.505199322 and -1.505199322.Subproblem 1
w + 0.875 = 1.505199322 Simplifying w + 0.875 = 1.505199322 Reorder the terms: 0.875 + w = 1.505199322 Solving 0.875 + w = 1.505199322 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = 1.505199322 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = 1.505199322 + -0.875 w = 1.505199322 + -0.875 Combine like terms: 1.505199322 + -0.875 = 0.630199322 w = 0.630199322 Simplifying w = 0.630199322Subproblem 2
w + 0.875 = -1.505199322 Simplifying w + 0.875 = -1.505199322 Reorder the terms: 0.875 + w = -1.505199322 Solving 0.875 + w = -1.505199322 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = -1.505199322 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = -1.505199322 + -0.875 w = -1.505199322 + -0.875 Combine like terms: -1.505199322 + -0.875 = -2.380199322 w = -2.380199322 Simplifying w = -2.380199322Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.630199322, -2.380199322}
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