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Simplifying (-3w2 + -3w + 2) + -1(-3w2 + 6w + 7) + (-5w2 + 8w + 6) = 0 Reorder the terms: (2 + -3w + -3w2) + -1(-3w2 + 6w + 7) + (-5w2 + 8w + 6) = 0 Remove parenthesis around (2 + -3w + -3w2) 2 + -3w + -3w2 + -1(-3w2 + 6w + 7) + (-5w2 + 8w + 6) = 0 Reorder the terms: 2 + -3w + -3w2 + -1(7 + 6w + -3w2) + (-5w2 + 8w + 6) = 0 2 + -3w + -3w2 + (7 * -1 + 6w * -1 + -3w2 * -1) + (-5w2 + 8w + 6) = 0 2 + -3w + -3w2 + (-7 + -6w + 3w2) + (-5w2 + 8w + 6) = 0 Reorder the terms: 2 + -3w + -3w2 + -7 + -6w + 3w2 + (6 + 8w + -5w2) = 0 Remove parenthesis around (6 + 8w + -5w2) 2 + -3w + -3w2 + -7 + -6w + 3w2 + 6 + 8w + -5w2 = 0 Reorder the terms: 2 + -7 + 6 + -3w + -6w + 8w + -3w2 + 3w2 + -5w2 = 0 Combine like terms: 2 + -7 = -5 -5 + 6 + -3w + -6w + 8w + -3w2 + 3w2 + -5w2 = 0 Combine like terms: -5 + 6 = 1 1 + -3w + -6w + 8w + -3w2 + 3w2 + -5w2 = 0 Combine like terms: -3w + -6w = -9w 1 + -9w + 8w + -3w2 + 3w2 + -5w2 = 0 Combine like terms: -9w + 8w = -1w 1 + -1w + -3w2 + 3w2 + -5w2 = 0 Combine like terms: -3w2 + 3w2 = 0 1 + -1w + 0 + -5w2 = 0 1 + -1w + -5w2 = 0 Solving 1 + -1w + -5w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.2 + 0.2w + w2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.2w + 0.2 + w2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.2w + w2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.2w + w2 = 0 + 0.2 0.2w + w2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 0.2w + w2 = 0.2 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 = 0.2 + 0.01 Reorder the terms: 0.01 + 0.2w + w2 = 0.2 + 0.01 Combine like terms: 0.2 + 0.01 = 0.21 0.01 + 0.2w + w2 = 0.21 Factor a perfect square on the left side: (w + 0.1)(w + 0.1) = 0.21 Calculate the square root of the right side: 0.458257569 Break this problem into two subproblems by setting (w + 0.1) equal to 0.458257569 and -0.458257569.Subproblem 1
w + 0.1 = 0.458257569 Simplifying w + 0.1 = 0.458257569 Reorder the terms: 0.1 + w = 0.458257569 Solving 0.1 + w = 0.458257569 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 = 0.458257569 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + w = 0.458257569 + -0.1 w = 0.458257569 + -0.1 Combine like terms: 0.458257569 + -0.1 = 0.358257569 w = 0.358257569 Simplifying w = 0.358257569Subproblem 2
w + 0.1 = -0.458257569 Simplifying w + 0.1 = -0.458257569 Reorder the terms: 0.1 + w = -0.458257569 Solving 0.1 + w = -0.458257569 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 = -0.458257569 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + w = -0.458257569 + -0.1 w = -0.458257569 + -0.1 Combine like terms: -0.458257569 + -0.1 = -0.558257569 w = -0.558257569 Simplifying w = -0.558257569Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.358257569, -0.558257569}
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