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Simplifying w2 = 3w + -1 Reorder the terms: w2 = -1 + 3w Solving w2 = -1 + 3w Solving for variable 'w'. Reorder the terms: 1 + -3w + w2 = -1 + 3w + 1 + -3w Reorder the terms: 1 + -3w + w2 = -1 + 1 + 3w + -3w Combine like terms: -1 + 1 = 0 1 + -3w + w2 = 0 + 3w + -3w 1 + -3w + w2 = 3w + -3w Combine like terms: 3w + -3w = 0 1 + -3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3w + -1 + w2 = 0 + -1 Reorder the terms: 1 + -1 + -3w + w2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3w + w2 = 0 + -1 -3w + w2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3w + w2 = -1 The w term is -3w. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3w + 2.25 + w2 = -1 + 2.25 Reorder the terms: 2.25 + -3w + w2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3w + w2 = 1.25 Factor a perfect square on the left side: (w + -1.5)(w + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (w + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
w + -1.5 = 1.118033989 Simplifying w + -1.5 = 1.118033989 Reorder the terms: -1.5 + w = 1.118033989 Solving -1.5 + w = 1.118033989 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + w = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + w = 1.118033989 + 1.5 w = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 w = 2.618033989 Simplifying w = 2.618033989Subproblem 2
w + -1.5 = -1.118033989 Simplifying w + -1.5 = -1.118033989 Reorder the terms: -1.5 + w = -1.118033989 Solving -1.5 + w = -1.118033989 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + w = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + w = -1.118033989 + 1.5 w = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 w = 0.381966011 Simplifying w = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.618033989, 0.381966011}
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