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Simplifying 5w2 + -4w + 1 = 3w Reorder the terms: 1 + -4w + 5w2 = 3w Solving 1 + -4w + 5w2 = 3w Solving for variable 'w'. Reorder the terms: 1 + -4w + -3w + 5w2 = 3w + -3w Combine like terms: -4w + -3w = -7w 1 + -7w + 5w2 = 3w + -3w Combine like terms: 3w + -3w = 0 1 + -7w + 5w2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + -1.4w + w2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -1.4w + -0.2 + w2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -1.4w + w2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -1.4w + w2 = 0 + -0.2 -1.4w + w2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -1.4w + w2 = -0.2 The w term is -1.4w. Take half its coefficient (-0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. -1.4w + 0.49 + w2 = -0.2 + 0.49 Reorder the terms: 0.49 + -1.4w + w2 = -0.2 + 0.49 Combine like terms: -0.2 + 0.49 = 0.29 0.49 + -1.4w + w2 = 0.29 Factor a perfect square on the left side: (w + -0.7)(w + -0.7) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (w + -0.7) equal to 0.538516481 and -0.538516481.Subproblem 1
w + -0.7 = 0.538516481 Simplifying w + -0.7 = 0.538516481 Reorder the terms: -0.7 + w = 0.538516481 Solving -0.7 + w = 0.538516481 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + w = 0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + w = 0.538516481 + 0.7 w = 0.538516481 + 0.7 Combine like terms: 0.538516481 + 0.7 = 1.238516481 w = 1.238516481 Simplifying w = 1.238516481Subproblem 2
w + -0.7 = -0.538516481 Simplifying w + -0.7 = -0.538516481 Reorder the terms: -0.7 + w = -0.538516481 Solving -0.7 + w = -0.538516481 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + w = -0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + w = -0.538516481 + 0.7 w = -0.538516481 + 0.7 Combine like terms: -0.538516481 + 0.7 = 0.161483519 w = 0.161483519 Simplifying w = 0.161483519Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.238516481, 0.161483519}
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