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Simplifying 2w2 + 6w + 13 = w2 + 10w + 12 Reorder the terms: 13 + 6w + 2w2 = w2 + 10w + 12 Reorder the terms: 13 + 6w + 2w2 = 12 + 10w + w2 Solving 13 + 6w + 2w2 = 12 + 10w + w2 Solving for variable 'w'. Reorder the terms: 13 + -12 + 6w + -10w + 2w2 + -1w2 = 12 + 10w + w2 + -12 + -10w + -1w2 Combine like terms: 13 + -12 = 1 1 + 6w + -10w + 2w2 + -1w2 = 12 + 10w + w2 + -12 + -10w + -1w2 Combine like terms: 6w + -10w = -4w 1 + -4w + 2w2 + -1w2 = 12 + 10w + w2 + -12 + -10w + -1w2 Combine like terms: 2w2 + -1w2 = 1w2 1 + -4w + 1w2 = 12 + 10w + w2 + -12 + -10w + -1w2 Reorder the terms: 1 + -4w + 1w2 = 12 + -12 + 10w + -10w + w2 + -1w2 Combine like terms: 12 + -12 = 0 1 + -4w + 1w2 = 0 + 10w + -10w + w2 + -1w2 1 + -4w + 1w2 = 10w + -10w + w2 + -1w2 Combine like terms: 10w + -10w = 0 1 + -4w + 1w2 = 0 + w2 + -1w2 1 + -4w + 1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 1 + -4w + 1w2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4w + -1 + w2 = 0 + -1 Reorder the terms: 1 + -1 + -4w + w2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4w + w2 = 0 + -1 -4w + w2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4w + w2 = -1 The w term is -4w. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4w + 4 + w2 = -1 + 4 Reorder the terms: 4 + -4w + w2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4w + w2 = 3 Factor a perfect square on the left side: (w + -2)(w + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (w + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
w + -2 = 1.732050808 Simplifying w + -2 = 1.732050808 Reorder the terms: -2 + w = 1.732050808 Solving -2 + w = 1.732050808 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + w = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + w = 1.732050808 + 2 w = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 w = 3.732050808 Simplifying w = 3.732050808Subproblem 2
w + -2 = -1.732050808 Simplifying w + -2 = -1.732050808 Reorder the terms: -2 + w = -1.732050808 Solving -2 + w = -1.732050808 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + w = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + w = -1.732050808 + 2 w = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 w = 0.267949192 Simplifying w = 0.267949192Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.732050808, 0.267949192}
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