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Simplifying 10w2 + -5w + -3 = 0 Reorder the terms: -3 + -5w + 10w2 = 0 Solving -3 + -5w + 10w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.3 + -0.5w + w2 = 0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + -0.5w + 0.3 + w2 = 0 + 0.3 Reorder the terms: -0.3 + 0.3 + -0.5w + w2 = 0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + -0.5w + w2 = 0 + 0.3 -0.5w + w2 = 0 + 0.3 Combine like terms: 0 + 0.3 = 0.3 -0.5w + w2 = 0.3 The w term is -0.5w. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5w + 0.0625 + w2 = 0.3 + 0.0625 Reorder the terms: 0.0625 + -0.5w + w2 = 0.3 + 0.0625 Combine like terms: 0.3 + 0.0625 = 0.3625 0.0625 + -0.5w + w2 = 0.3625 Factor a perfect square on the left side: (w + -0.25)(w + -0.25) = 0.3625 Calculate the square root of the right side: 0.602079729 Break this problem into two subproblems by setting (w + -0.25) equal to 0.602079729 and -0.602079729.Subproblem 1
w + -0.25 = 0.602079729 Simplifying w + -0.25 = 0.602079729 Reorder the terms: -0.25 + w = 0.602079729 Solving -0.25 + w = 0.602079729 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = 0.602079729 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = 0.602079729 + 0.25 w = 0.602079729 + 0.25 Combine like terms: 0.602079729 + 0.25 = 0.852079729 w = 0.852079729 Simplifying w = 0.852079729Subproblem 2
w + -0.25 = -0.602079729 Simplifying w + -0.25 = -0.602079729 Reorder the terms: -0.25 + w = -0.602079729 Solving -0.25 + w = -0.602079729 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = -0.602079729 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = -0.602079729 + 0.25 w = -0.602079729 + 0.25 Combine like terms: -0.602079729 + 0.25 = -0.352079729 w = -0.352079729 Simplifying w = -0.352079729Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.852079729, -0.352079729}
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