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Simplifying 20 = 8(w + 7) * 4w Reorder the terms: 20 = 8(7 + w) * 4w Reorder the terms for easier multiplication: 20 = 8 * 4w(7 + w) Multiply 8 * 4 20 = 32w(7 + w) 20 = (7 * 32w + w * 32w) 20 = (224w + 32w2) Solving 20 = 224w + 32w2 Solving for variable 'w'. Reorder the terms: 20 + -224w + -32w2 = 224w + -224w + 32w2 + -32w2 Combine like terms: 224w + -224w = 0 20 + -224w + -32w2 = 0 + 32w2 + -32w2 20 + -224w + -32w2 = 32w2 + -32w2 Combine like terms: 32w2 + -32w2 = 0 20 + -224w + -32w2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(5 + -56w + -8w2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(5 + -56w + -8w2)' equal to zero and attempt to solve: Simplifying 5 + -56w + -8w2 = 0 Solving 5 + -56w + -8w2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.625 + 7w + w2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 7w + 0.625 + w2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 7w + w2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 7w + w2 = 0 + 0.625 7w + w2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 7w + w2 = 0.625 The w term is 7w. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7w + 12.25 + w2 = 0.625 + 12.25 Reorder the terms: 12.25 + 7w + w2 = 0.625 + 12.25 Combine like terms: 0.625 + 12.25 = 12.875 12.25 + 7w + w2 = 12.875 Factor a perfect square on the left side: (w + 3.5)(w + 3.5) = 12.875 Calculate the square root of the right side: 3.588175024 Break this problem into two subproblems by setting (w + 3.5) equal to 3.588175024 and -3.588175024.Subproblem 1
w + 3.5 = 3.588175024 Simplifying w + 3.5 = 3.588175024 Reorder the terms: 3.5 + w = 3.588175024 Solving 3.5 + w = 3.588175024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + w = 3.588175024 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + w = 3.588175024 + -3.5 w = 3.588175024 + -3.5 Combine like terms: 3.588175024 + -3.5 = 0.088175024 w = 0.088175024 Simplifying w = 0.088175024Subproblem 2
w + 3.5 = -3.588175024 Simplifying w + 3.5 = -3.588175024 Reorder the terms: 3.5 + w = -3.588175024 Solving 3.5 + w = -3.588175024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + w = -3.588175024 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + w = -3.588175024 + -3.5 w = -3.588175024 + -3.5 Combine like terms: -3.588175024 + -3.5 = -7.088175024 w = -7.088175024 Simplifying w = -7.088175024Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.088175024, -7.088175024}Solution
w = {0.088175024, -7.088175024}
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