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Simplifying 20 + -4(2w + 1) = 9w(9 + w) Reorder the terms: 20 + -4(1 + 2w) = 9w(9 + w) 20 + (1 * -4 + 2w * -4) = 9w(9 + w) 20 + (-4 + -8w) = 9w(9 + w) Combine like terms: 20 + -4 = 16 16 + -8w = 9w(9 + w) 16 + -8w = (9 * 9w + w * 9w) 16 + -8w = (81w + 9w2) Solving 16 + -8w = 81w + 9w2 Solving for variable 'w'. Combine like terms: -8w + -81w = -89w 16 + -89w + -9w2 = 81w + 9w2 + -81w + -9w2 Reorder the terms: 16 + -89w + -9w2 = 81w + -81w + 9w2 + -9w2 Combine like terms: 81w + -81w = 0 16 + -89w + -9w2 = 0 + 9w2 + -9w2 16 + -89w + -9w2 = 9w2 + -9w2 Combine like terms: 9w2 + -9w2 = 0 16 + -89w + -9w2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -1.777777778 + 9.888888889w + w2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 9.888888889w + 1.777777778 + w2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 9.888888889w + w2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 9.888888889w + w2 = 0 + 1.777777778 9.888888889w + w2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 9.888888889w + w2 = 1.777777778 The w term is 9.888888889w. Take half its coefficient (4.944444445). Square it (24.44753087) and add it to both sides. Add '24.44753087' to each side of the equation. 9.888888889w + 24.44753087 + w2 = 1.777777778 + 24.44753087 Reorder the terms: 24.44753087 + 9.888888889w + w2 = 1.777777778 + 24.44753087 Combine like terms: 1.777777778 + 24.44753087 = 26.225308648 24.44753087 + 9.888888889w + w2 = 26.225308648 Factor a perfect square on the left side: (w + 4.944444445)(w + 4.944444445) = 26.225308648 Calculate the square root of the right side: 5.121065187 Break this problem into two subproblems by setting (w + 4.944444445) equal to 5.121065187 and -5.121065187.Subproblem 1
w + 4.944444445 = 5.121065187 Simplifying w + 4.944444445 = 5.121065187 Reorder the terms: 4.944444445 + w = 5.121065187 Solving 4.944444445 + w = 5.121065187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.944444445' to each side of the equation. 4.944444445 + -4.944444445 + w = 5.121065187 + -4.944444445 Combine like terms: 4.944444445 + -4.944444445 = 0.000000000 0.000000000 + w = 5.121065187 + -4.944444445 w = 5.121065187 + -4.944444445 Combine like terms: 5.121065187 + -4.944444445 = 0.176620742 w = 0.176620742 Simplifying w = 0.176620742Subproblem 2
w + 4.944444445 = -5.121065187 Simplifying w + 4.944444445 = -5.121065187 Reorder the terms: 4.944444445 + w = -5.121065187 Solving 4.944444445 + w = -5.121065187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.944444445' to each side of the equation. 4.944444445 + -4.944444445 + w = -5.121065187 + -4.944444445 Combine like terms: 4.944444445 + -4.944444445 = 0.000000000 0.000000000 + w = -5.121065187 + -4.944444445 w = -5.121065187 + -4.944444445 Combine like terms: -5.121065187 + -4.944444445 = -10.065509632 w = -10.065509632 Simplifying w = -10.065509632Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.176620742, -10.065509632}
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