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Simplifying (w + 5)(2w + 5) = 5 Reorder the terms: (5 + w)(2w + 5) = 5 Reorder the terms: (5 + w)(5 + 2w) = 5 Multiply (5 + w) * (5 + 2w) (5(5 + 2w) + w(5 + 2w)) = 5 ((5 * 5 + 2w * 5) + w(5 + 2w)) = 5 ((25 + 10w) + w(5 + 2w)) = 5 (25 + 10w + (5 * w + 2w * w)) = 5 (25 + 10w + (5w + 2w2)) = 5 Combine like terms: 10w + 5w = 15w (25 + 15w + 2w2) = 5 Solving 25 + 15w + 2w2 = 5 Solving for variable 'w'. Reorder the terms: 25 + -5 + 15w + 2w2 = 5 + -5 Combine like terms: 25 + -5 = 20 20 + 15w + 2w2 = 5 + -5 Combine like terms: 5 + -5 = 0 20 + 15w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 10 + 7.5w + w2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + 7.5w + -10 + w2 = 0 + -10 Reorder the terms: 10 + -10 + 7.5w + w2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 7.5w + w2 = 0 + -10 7.5w + w2 = 0 + -10 Combine like terms: 0 + -10 = -10 7.5w + w2 = -10 The w term is 7.5w. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5w + 14.0625 + w2 = -10 + 14.0625 Reorder the terms: 14.0625 + 7.5w + w2 = -10 + 14.0625 Combine like terms: -10 + 14.0625 = 4.0625 14.0625 + 7.5w + w2 = 4.0625 Factor a perfect square on the left side: (w + 3.75)(w + 3.75) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (w + 3.75) equal to 2.015564437 and -2.015564437.Subproblem 1
w + 3.75 = 2.015564437 Simplifying w + 3.75 = 2.015564437 Reorder the terms: 3.75 + w = 2.015564437 Solving 3.75 + w = 2.015564437 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + w = 2.015564437 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + w = 2.015564437 + -3.75 w = 2.015564437 + -3.75 Combine like terms: 2.015564437 + -3.75 = -1.734435563 w = -1.734435563 Simplifying w = -1.734435563Subproblem 2
w + 3.75 = -2.015564437 Simplifying w + 3.75 = -2.015564437 Reorder the terms: 3.75 + w = -2.015564437 Solving 3.75 + w = -2.015564437 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + w = -2.015564437 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + w = -2.015564437 + -3.75 w = -2.015564437 + -3.75 Combine like terms: -2.015564437 + -3.75 = -5.765564437 w = -5.765564437 Simplifying w = -5.765564437Solution
The solution to the problem is based on the solutions from the subproblems. w = {-1.734435563, -5.765564437}
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