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Simplifying 10w + 19 + 3w = 6 + w(9 + w) + -14 Reorder the terms: 19 + 10w + 3w = 6 + w(9 + w) + -14 Combine like terms: 10w + 3w = 13w 19 + 13w = 6 + w(9 + w) + -14 19 + 13w = 6 + (9 * w + w * w) + -14 19 + 13w = 6 + (9w + w2) + -14 Reorder the terms: 19 + 13w = 6 + -14 + 9w + w2 Combine like terms: 6 + -14 = -8 19 + 13w = -8 + 9w + w2 Solving 19 + 13w = -8 + 9w + w2 Solving for variable 'w'. Reorder the terms: 19 + 8 + 13w + -9w + -1w2 = -8 + 9w + w2 + 8 + -9w + -1w2 Combine like terms: 19 + 8 = 27 27 + 13w + -9w + -1w2 = -8 + 9w + w2 + 8 + -9w + -1w2 Combine like terms: 13w + -9w = 4w 27 + 4w + -1w2 = -8 + 9w + w2 + 8 + -9w + -1w2 Reorder the terms: 27 + 4w + -1w2 = -8 + 8 + 9w + -9w + w2 + -1w2 Combine like terms: -8 + 8 = 0 27 + 4w + -1w2 = 0 + 9w + -9w + w2 + -1w2 27 + 4w + -1w2 = 9w + -9w + w2 + -1w2 Combine like terms: 9w + -9w = 0 27 + 4w + -1w2 = 0 + w2 + -1w2 27 + 4w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 27 + 4w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -27 + -4w + w2 = 0 Move the constant term to the right: Add '27' to each side of the equation. -27 + -4w + 27 + w2 = 0 + 27 Reorder the terms: -27 + 27 + -4w + w2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + -4w + w2 = 0 + 27 -4w + w2 = 0 + 27 Combine like terms: 0 + 27 = 27 -4w + w2 = 27 The w term is -4w. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4w + 4 + w2 = 27 + 4 Reorder the terms: 4 + -4w + w2 = 27 + 4 Combine like terms: 27 + 4 = 31 4 + -4w + w2 = 31 Factor a perfect square on the left side: (w + -2)(w + -2) = 31 Calculate the square root of the right side: 5.567764363 Break this problem into two subproblems by setting (w + -2) equal to 5.567764363 and -5.567764363.Subproblem 1
w + -2 = 5.567764363 Simplifying w + -2 = 5.567764363 Reorder the terms: -2 + w = 5.567764363 Solving -2 + w = 5.567764363 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + w = 5.567764363 + 2 Combine like terms: -2 + 2 = 0 0 + w = 5.567764363 + 2 w = 5.567764363 + 2 Combine like terms: 5.567764363 + 2 = 7.567764363 w = 7.567764363 Simplifying w = 7.567764363Subproblem 2
w + -2 = -5.567764363 Simplifying w + -2 = -5.567764363 Reorder the terms: -2 + w = -5.567764363 Solving -2 + w = -5.567764363 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + w = -5.567764363 + 2 Combine like terms: -2 + 2 = 0 0 + w = -5.567764363 + 2 w = -5.567764363 + 2 Combine like terms: -5.567764363 + 2 = -3.567764363 w = -3.567764363 Simplifying w = -3.567764363Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.567764363, -3.567764363}
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