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Simplifying 3w + -5 + -19w2 + 17 = 0 Reorder the terms: -5 + 17 + 3w + -19w2 = 0 Combine like terms: -5 + 17 = 12 12 + 3w + -19w2 = 0 Solving 12 + 3w + -19w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -19 the coefficient of the squared term: Divide each side by '-19'. -0.6315789474 + -0.1578947368w + w2 = 0 Move the constant term to the right: Add '0.6315789474' to each side of the equation. -0.6315789474 + -0.1578947368w + 0.6315789474 + w2 = 0 + 0.6315789474 Reorder the terms: -0.6315789474 + 0.6315789474 + -0.1578947368w + w2 = 0 + 0.6315789474 Combine like terms: -0.6315789474 + 0.6315789474 = 0.0000000000 0.0000000000 + -0.1578947368w + w2 = 0 + 0.6315789474 -0.1578947368w + w2 = 0 + 0.6315789474 Combine like terms: 0 + 0.6315789474 = 0.6315789474 -0.1578947368w + w2 = 0.6315789474 The w term is -0.1578947368w. Take half its coefficient (-0.0789473684). Square it (0.006232686977) and add it to both sides. Add '0.006232686977' to each side of the equation. -0.1578947368w + 0.006232686977 + w2 = 0.6315789474 + 0.006232686977 Reorder the terms: 0.006232686977 + -0.1578947368w + w2 = 0.6315789474 + 0.006232686977 Combine like terms: 0.6315789474 + 0.006232686977 = 0.637811634377 0.006232686977 + -0.1578947368w + w2 = 0.637811634377 Factor a perfect square on the left side: (w + -0.0789473684)(w + -0.0789473684) = 0.637811634377 Calculate the square root of the right side: 0.7986311 Break this problem into two subproblems by setting (w + -0.0789473684) equal to 0.7986311 and -0.7986311.Subproblem 1
w + -0.0789473684 = 0.7986311 Simplifying w + -0.0789473684 = 0.7986311 Reorder the terms: -0.0789473684 + w = 0.7986311 Solving -0.0789473684 + w = 0.7986311 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0789473684' to each side of the equation. -0.0789473684 + 0.0789473684 + w = 0.7986311 + 0.0789473684 Combine like terms: -0.0789473684 + 0.0789473684 = 0.0000000000 0.0000000000 + w = 0.7986311 + 0.0789473684 w = 0.7986311 + 0.0789473684 Combine like terms: 0.7986311 + 0.0789473684 = 0.8775784684 w = 0.8775784684 Simplifying w = 0.8775784684Subproblem 2
w + -0.0789473684 = -0.7986311 Simplifying w + -0.0789473684 = -0.7986311 Reorder the terms: -0.0789473684 + w = -0.7986311 Solving -0.0789473684 + w = -0.7986311 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0789473684' to each side of the equation. -0.0789473684 + 0.0789473684 + w = -0.7986311 + 0.0789473684 Combine like terms: -0.0789473684 + 0.0789473684 = 0.0000000000 0.0000000000 + w = -0.7986311 + 0.0789473684 w = -0.7986311 + 0.0789473684 Combine like terms: -0.7986311 + 0.0789473684 = -0.7196837316 w = -0.7196837316 Simplifying w = -0.7196837316Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.8775784684, -0.7196837316}
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