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Simplifying 4w2 + 12w = 5 Reorder the terms: 12w + 4w2 = 5 Solving 12w + 4w2 = 5 Solving for variable 'w'. Reorder the terms: -5 + 12w + 4w2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 12w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 3w + w2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 3w + 1.25 + w2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 3w + w2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 3w + w2 = 0 + 1.25 3w + w2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 3w + w2 = 1.25 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 = 1.25 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 1.25 + 2.25 Combine like terms: 1.25 + 2.25 = 3.5 2.25 + 3w + w2 = 3.5 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (w + 1.5) equal to 1.870828693 and -1.870828693.Subproblem 1
w + 1.5 = 1.870828693 Simplifying w + 1.5 = 1.870828693 Reorder the terms: 1.5 + w = 1.870828693 Solving 1.5 + w = 1.870828693 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 = 1.870828693 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 1.870828693 + -1.5 w = 1.870828693 + -1.5 Combine like terms: 1.870828693 + -1.5 = 0.370828693 w = 0.370828693 Simplifying w = 0.370828693Subproblem 2
w + 1.5 = -1.870828693 Simplifying w + 1.5 = -1.870828693 Reorder the terms: 1.5 + w = -1.870828693 Solving 1.5 + w = -1.870828693 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 = -1.870828693 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -1.870828693 + -1.5 w = -1.870828693 + -1.5 Combine like terms: -1.870828693 + -1.5 = -3.370828693 w = -3.370828693 Simplifying w = -3.370828693Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.370828693, -3.370828693}
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