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Simplifying (40w + 200)(w + 2) = 40 Reorder the terms: (200 + 40w)(w + 2) = 40 Reorder the terms: (200 + 40w)(2 + w) = 40 Multiply (200 + 40w) * (2 + w) (200(2 + w) + 40w * (2 + w)) = 40 ((2 * 200 + w * 200) + 40w * (2 + w)) = 40 ((400 + 200w) + 40w * (2 + w)) = 40 (400 + 200w + (2 * 40w + w * 40w)) = 40 (400 + 200w + (80w + 40w2)) = 40 Combine like terms: 200w + 80w = 280w (400 + 280w + 40w2) = 40 Solving 400 + 280w + 40w2 = 40 Solving for variable 'w'. Reorder the terms: 400 + -40 + 280w + 40w2 = 40 + -40 Combine like terms: 400 + -40 = 360 360 + 280w + 40w2 = 40 + -40 Combine like terms: 40 + -40 = 0 360 + 280w + 40w2 = 0 Factor out the Greatest Common Factor (GCF), '40'. 40(9 + 7w + w2) = 0 Ignore the factor 40.Subproblem 1
Set the factor '(9 + 7w + w2)' equal to zero and attempt to solve: Simplifying 9 + 7w + w2 = 0 Solving 9 + 7w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '-9' to each side of the equation. 9 + 7w + -9 + w2 = 0 + -9 Reorder the terms: 9 + -9 + 7w + w2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + 7w + w2 = 0 + -9 7w + w2 = 0 + -9 Combine like terms: 0 + -9 = -9 7w + w2 = -9 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 = -9 + 12.25 Reorder the terms: 12.25 + 7w + w2 = -9 + 12.25 Combine like terms: -9 + 12.25 = 3.25 12.25 + 7w + w2 = 3.25 Factor a perfect square on the left side: (w + 3.5)(w + 3.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (w + 3.5) equal to 1.802775638 and -1.802775638.Subproblem 1
w + 3.5 = 1.802775638 Simplifying w + 3.5 = 1.802775638 Reorder the terms: 3.5 + w = 1.802775638 Solving 3.5 + w = 1.802775638 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 = 1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + w = 1.802775638 + -3.5 w = 1.802775638 + -3.5 Combine like terms: 1.802775638 + -3.5 = -1.697224362 w = -1.697224362 Simplifying w = -1.697224362Subproblem 2
w + 3.5 = -1.802775638 Simplifying w + 3.5 = -1.802775638 Reorder the terms: 3.5 + w = -1.802775638 Solving 3.5 + w = -1.802775638 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 = -1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + w = -1.802775638 + -3.5 w = -1.802775638 + -3.5 Combine like terms: -1.802775638 + -3.5 = -5.302775638 w = -5.302775638 Simplifying w = -5.302775638Solution
The solution to the problem is based on the solutions from the subproblems. w = {-1.697224362, -5.302775638}Solution
w = {-1.697224362, -5.302775638}
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