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Simplifying 2w2 + 4w + -170 = w2 + 3w Reorder the terms: -170 + 4w + 2w2 = w2 + 3w Reorder the terms: -170 + 4w + 2w2 = 3w + w2 Solving -170 + 4w + 2w2 = 3w + w2 Solving for variable 'w'. Reorder the terms: -170 + 4w + -3w + 2w2 + -1w2 = 3w + w2 + -3w + -1w2 Combine like terms: 4w + -3w = 1w -170 + 1w + 2w2 + -1w2 = 3w + w2 + -3w + -1w2 Combine like terms: 2w2 + -1w2 = 1w2 -170 + 1w + 1w2 = 3w + w2 + -3w + -1w2 Reorder the terms: -170 + 1w + 1w2 = 3w + -3w + w2 + -1w2 Combine like terms: 3w + -3w = 0 -170 + 1w + 1w2 = 0 + w2 + -1w2 -170 + 1w + 1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 -170 + 1w + 1w2 = 0 Begin completing the square. Move the constant term to the right: Add '170' to each side of the equation. -170 + 1w + 170 + w2 = 0 + 170 Reorder the terms: -170 + 170 + 1w + w2 = 0 + 170 Combine like terms: -170 + 170 = 0 0 + 1w + w2 = 0 + 170 1w + w2 = 0 + 170 Combine like terms: 0 + 170 = 170 1w + w2 = 170 The w term is 1w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. w + 0.25 + w2 = 170 + 0.25 Reorder the terms: 0.25 + w + w2 = 170 + 0.25 Combine like terms: 170 + 0.25 = 170.25 0.25 + w + w2 = 170.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 170.25 Calculate the square root of the right side: 13.047988351 Break this problem into two subproblems by setting (w + 0.5) equal to 13.047988351 and -13.047988351.Subproblem 1
w + 0.5 = 13.047988351 Simplifying w + 0.5 = 13.047988351 Reorder the terms: 0.5 + w = 13.047988351 Solving 0.5 + w = 13.047988351 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 13.047988351 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 13.047988351 + -0.5 w = 13.047988351 + -0.5 Combine like terms: 13.047988351 + -0.5 = 12.547988351 w = 12.547988351 Simplifying w = 12.547988351Subproblem 2
w + 0.5 = -13.047988351 Simplifying w + 0.5 = -13.047988351 Reorder the terms: 0.5 + w = -13.047988351 Solving 0.5 + w = -13.047988351 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -13.047988351 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -13.047988351 + -0.5 w = -13.047988351 + -0.5 Combine like terms: -13.047988351 + -0.5 = -13.547988351 w = -13.547988351 Simplifying w = -13.547988351Solution
The solution to the problem is based on the solutions from the subproblems. w = {12.547988351, -13.547988351}
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