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Simplifying (10w + -1)(10w + 1) = 20w Reorder the terms: (-1 + 10w)(10w + 1) = 20w Reorder the terms: (-1 + 10w)(1 + 10w) = 20w Multiply (-1 + 10w) * (1 + 10w) (-1(1 + 10w) + 10w * (1 + 10w)) = 20w ((1 * -1 + 10w * -1) + 10w * (1 + 10w)) = 20w ((-1 + -10w) + 10w * (1 + 10w)) = 20w (-1 + -10w + (1 * 10w + 10w * 10w)) = 20w (-1 + -10w + (10w + 100w2)) = 20w Combine like terms: -10w + 10w = 0 (-1 + 0 + 100w2) = 20w (-1 + 100w2) = 20w Solving -1 + 100w2 = 20w Solving for variable 'w'. Reorder the terms: -1 + -20w + 100w2 = 20w + -20w Combine like terms: 20w + -20w = 0 -1 + -20w + 100w2 = 0 Begin completing the square. Divide all terms by 100 the coefficient of the squared term: Divide each side by '100'. -0.01 + -0.2w + w2 = 0.00 Move the constant term to the right: Add '0.01' to each side of the equation. -0.01 + -0.2w + 0.01 + w2 = 0.00 + 0.01 Reorder the terms: -0.01 + 0.01 + -0.2w + w2 = 0.00 + 0.01 Combine like terms: -0.01 + 0.01 = 0.00 0.00 + -0.2w + w2 = 0.00 + 0.01 -0.2w + w2 = 0.00 + 0.01 Combine like terms: 0.00 + 0.01 = 0.01 -0.2w + w2 = 0.01 The w term is -0.2w. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2w + 0.01 + w2 = 0.01 + 0.01 Reorder the terms: 0.01 + -0.2w + w2 = 0.01 + 0.01 Combine like terms: 0.01 + 0.01 = 0.02 0.01 + -0.2w + w2 = 0.02 Factor a perfect square on the left side: (w + -0.1)(w + -0.1) = 0.02 Calculate the square root of the right side: 0.141421356 Break this problem into two subproblems by setting (w + -0.1) equal to 0.141421356 and -0.141421356.Subproblem 1
w + -0.1 = 0.141421356 Simplifying w + -0.1 = 0.141421356 Reorder the terms: -0.1 + w = 0.141421356 Solving -0.1 + w = 0.141421356 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + w = 0.141421356 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + w = 0.141421356 + 0.1 w = 0.141421356 + 0.1 Combine like terms: 0.141421356 + 0.1 = 0.241421356 w = 0.241421356 Simplifying w = 0.241421356Subproblem 2
w + -0.1 = -0.141421356 Simplifying w + -0.1 = -0.141421356 Reorder the terms: -0.1 + w = -0.141421356 Solving -0.1 + w = -0.141421356 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + w = -0.141421356 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + w = -0.141421356 + 0.1 w = -0.141421356 + 0.1 Combine like terms: -0.141421356 + 0.1 = -0.041421356 w = -0.041421356 Simplifying w = -0.041421356Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.241421356, -0.041421356}
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