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Simplifying 0.036 = (w + 0.18) * w Reorder the terms: 0.036 = (0.18 + w) * w Reorder the terms for easier multiplication: 0.036 = w(0.18 + w) 0.036 = (0.18 * w + w * w) 0.036 = (0.18w + w2) Solving 0.036 = 0.18w + w2 Solving for variable 'w'. Reorder the terms: 0.036 + -0.18w + -1w2 = 0.18w + -0.18w + w2 + -1w2 Combine like terms: 0.18w + -0.18w = 0.00 0.036 + -0.18w + -1w2 = 0.00 + w2 + -1w2 0.036 + -0.18w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 0.036 + -0.18w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -0.036 + 0.18w + w2 = 0 Move the constant term to the right: Add '0.036' to each side of the equation. -0.036 + 0.18w + 0.036 + w2 = 0 + 0.036 Reorder the terms: -0.036 + 0.036 + 0.18w + w2 = 0 + 0.036 Combine like terms: -0.036 + 0.036 = 0.000 0.000 + 0.18w + w2 = 0 + 0.036 0.18w + w2 = 0 + 0.036 Combine like terms: 0 + 0.036 = 0.036 0.18w + w2 = 0.036 The w term is 0.18w. Take half its coefficient (0.09). Square it (0.0081) and add it to both sides. Add '0.0081' to each side of the equation. 0.18w + 0.0081 + w2 = 0.036 + 0.0081 Reorder the terms: 0.0081 + 0.18w + w2 = 0.036 + 0.0081 Combine like terms: 0.036 + 0.0081 = 0.0441 0.0081 + 0.18w + w2 = 0.0441 Factor a perfect square on the left side: (w + 0.09)(w + 0.09) = 0.0441 Calculate the square root of the right side: 0.21 Break this problem into two subproblems by setting (w + 0.09) equal to 0.21 and -0.21.Subproblem 1
w + 0.09 = 0.21 Simplifying w + 0.09 = 0.21 Reorder the terms: 0.09 + w = 0.21 Solving 0.09 + w = 0.21 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.09' to each side of the equation. 0.09 + -0.09 + w = 0.21 + -0.09 Combine like terms: 0.09 + -0.09 = 0.00 0.00 + w = 0.21 + -0.09 w = 0.21 + -0.09 Combine like terms: 0.21 + -0.09 = 0.12 w = 0.12 Simplifying w = 0.12Subproblem 2
w + 0.09 = -0.21 Simplifying w + 0.09 = -0.21 Reorder the terms: 0.09 + w = -0.21 Solving 0.09 + w = -0.21 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.09' to each side of the equation. 0.09 + -0.09 + w = -0.21 + -0.09 Combine like terms: 0.09 + -0.09 = 0.00 0.00 + w = -0.21 + -0.09 w = -0.21 + -0.09 Combine like terms: -0.21 + -0.09 = -0.3 w = -0.3 Simplifying w = -0.3Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.12, -0.3}
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