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Simplifying 2w2 + -9w + 5 = w2 + 6w + 9 Reorder the terms: 5 + -9w + 2w2 = w2 + 6w + 9 Reorder the terms: 5 + -9w + 2w2 = 9 + 6w + w2 Solving 5 + -9w + 2w2 = 9 + 6w + w2 Solving for variable 'w'. Reorder the terms: 5 + -9 + -9w + -6w + 2w2 + -1w2 = 9 + 6w + w2 + -9 + -6w + -1w2 Combine like terms: 5 + -9 = -4 -4 + -9w + -6w + 2w2 + -1w2 = 9 + 6w + w2 + -9 + -6w + -1w2 Combine like terms: -9w + -6w = -15w -4 + -15w + 2w2 + -1w2 = 9 + 6w + w2 + -9 + -6w + -1w2 Combine like terms: 2w2 + -1w2 = 1w2 -4 + -15w + 1w2 = 9 + 6w + w2 + -9 + -6w + -1w2 Reorder the terms: -4 + -15w + 1w2 = 9 + -9 + 6w + -6w + w2 + -1w2 Combine like terms: 9 + -9 = 0 -4 + -15w + 1w2 = 0 + 6w + -6w + w2 + -1w2 -4 + -15w + 1w2 = 6w + -6w + w2 + -1w2 Combine like terms: 6w + -6w = 0 -4 + -15w + 1w2 = 0 + w2 + -1w2 -4 + -15w + 1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 -4 + -15w + 1w2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + -15w + 4 + w2 = 0 + 4 Reorder the terms: -4 + 4 + -15w + w2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -15w + w2 = 0 + 4 -15w + w2 = 0 + 4 Combine like terms: 0 + 4 = 4 -15w + w2 = 4 The w term is -15w. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15w + 56.25 + w2 = 4 + 56.25 Reorder the terms: 56.25 + -15w + w2 = 4 + 56.25 Combine like terms: 4 + 56.25 = 60.25 56.25 + -15w + w2 = 60.25 Factor a perfect square on the left side: (w + -7.5)(w + -7.5) = 60.25 Calculate the square root of the right side: 7.762087348 Break this problem into two subproblems by setting (w + -7.5) equal to 7.762087348 and -7.762087348.Subproblem 1
w + -7.5 = 7.762087348 Simplifying w + -7.5 = 7.762087348 Reorder the terms: -7.5 + w = 7.762087348 Solving -7.5 + w = 7.762087348 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + w = 7.762087348 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + w = 7.762087348 + 7.5 w = 7.762087348 + 7.5 Combine like terms: 7.762087348 + 7.5 = 15.262087348 w = 15.262087348 Simplifying w = 15.262087348Subproblem 2
w + -7.5 = -7.762087348 Simplifying w + -7.5 = -7.762087348 Reorder the terms: -7.5 + w = -7.762087348 Solving -7.5 + w = -7.762087348 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + w = -7.762087348 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + w = -7.762087348 + 7.5 w = -7.762087348 + 7.5 Combine like terms: -7.762087348 + 7.5 = -0.262087348 w = -0.262087348 Simplifying w = -0.262087348Solution
The solution to the problem is based on the solutions from the subproblems. w = {15.262087348, -0.262087348}
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