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Simplifying (-3w2 + 7w + 7) + -1(7w2 + 11w + -4) = 0 Reorder the terms: (7 + 7w + -3w2) + -1(7w2 + 11w + -4) = 0 Remove parenthesis around (7 + 7w + -3w2) 7 + 7w + -3w2 + -1(7w2 + 11w + -4) = 0 Reorder the terms: 7 + 7w + -3w2 + -1(-4 + 11w + 7w2) = 0 7 + 7w + -3w2 + (-4 * -1 + 11w * -1 + 7w2 * -1) = 0 7 + 7w + -3w2 + (4 + -11w + -7w2) = 0 Reorder the terms: 7 + 4 + 7w + -11w + -3w2 + -7w2 = 0 Combine like terms: 7 + 4 = 11 11 + 7w + -11w + -3w2 + -7w2 = 0 Combine like terms: 7w + -11w = -4w 11 + -4w + -3w2 + -7w2 = 0 Combine like terms: -3w2 + -7w2 = -10w2 11 + -4w + -10w2 = 0 Solving 11 + -4w + -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'. -1.1 + 0.4w + w2 = 0.0 Move the constant term to the right: Add '1.1' to each side of the equation. -1.1 + 0.4w + 1.1 + w2 = 0.0 + 1.1 Reorder the terms: -1.1 + 1.1 + 0.4w + w2 = 0.0 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + 0.4w + w2 = 0.0 + 1.1 0.4w + w2 = 0.0 + 1.1 Combine like terms: 0.0 + 1.1 = 1.1 0.4w + w2 = 1.1 The w term is 0.4w. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4w + 0.04 + w2 = 1.1 + 0.04 Reorder the terms: 0.04 + 0.4w + w2 = 1.1 + 0.04 Combine like terms: 1.1 + 0.04 = 1.14 0.04 + 0.4w + w2 = 1.14 Factor a perfect square on the left side: (w + 0.2)(w + 0.2) = 1.14 Calculate the square root of the right side: 1.067707825 Break this problem into two subproblems by setting (w + 0.2) equal to 1.067707825 and -1.067707825.Subproblem 1
w + 0.2 = 1.067707825 Simplifying w + 0.2 = 1.067707825 Reorder the terms: 0.2 + w = 1.067707825 Solving 0.2 + w = 1.067707825 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + w = 1.067707825 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + w = 1.067707825 + -0.2 w = 1.067707825 + -0.2 Combine like terms: 1.067707825 + -0.2 = 0.867707825 w = 0.867707825 Simplifying w = 0.867707825Subproblem 2
w + 0.2 = -1.067707825 Simplifying w + 0.2 = -1.067707825 Reorder the terms: 0.2 + w = -1.067707825 Solving 0.2 + w = -1.067707825 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + w = -1.067707825 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + w = -1.067707825 + -0.2 w = -1.067707825 + -0.2 Combine like terms: -1.067707825 + -0.2 = -1.267707825 w = -1.267707825 Simplifying w = -1.267707825Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.867707825, -1.267707825}
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