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Simplifying (w + 3) = 2w2 + 4w + -15 Reorder the terms: (3 + w) = 2w2 + 4w + -15 Remove parenthesis around (3 + w) 3 + w = 2w2 + 4w + -15 Reorder the terms: 3 + w = -15 + 4w + 2w2 Solving 3 + w = -15 + 4w + 2w2 Solving for variable 'w'. Reorder the terms: 3 + 15 + w + -4w + -2w2 = -15 + 4w + 2w2 + 15 + -4w + -2w2 Combine like terms: 3 + 15 = 18 18 + w + -4w + -2w2 = -15 + 4w + 2w2 + 15 + -4w + -2w2 Combine like terms: w + -4w = -3w 18 + -3w + -2w2 = -15 + 4w + 2w2 + 15 + -4w + -2w2 Reorder the terms: 18 + -3w + -2w2 = -15 + 15 + 4w + -4w + 2w2 + -2w2 Combine like terms: -15 + 15 = 0 18 + -3w + -2w2 = 0 + 4w + -4w + 2w2 + -2w2 18 + -3w + -2w2 = 4w + -4w + 2w2 + -2w2 Combine like terms: 4w + -4w = 0 18 + -3w + -2w2 = 0 + 2w2 + -2w2 18 + -3w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 18 + -3w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -9 + 1.5w + w2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 1.5w + 9 + w2 = 0 + 9 Reorder the terms: -9 + 9 + 1.5w + w2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 1.5w + w2 = 0 + 9 1.5w + w2 = 0 + 9 Combine like terms: 0 + 9 = 9 1.5w + w2 = 9 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 9 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 9 + 0.5625 Combine like terms: 9 + 0.5625 = 9.5625 0.5625 + 1.5w + w2 = 9.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (w + 0.75) equal to 3.092329219 and -3.092329219.Subproblem 1
w + 0.75 = 3.092329219 Simplifying w + 0.75 = 3.092329219 Reorder the terms: 0.75 + w = 3.092329219 Solving 0.75 + w = 3.092329219 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 3.092329219 + -0.75 w = 3.092329219 + -0.75 Combine like terms: 3.092329219 + -0.75 = 2.342329219 w = 2.342329219 Simplifying w = 2.342329219Subproblem 2
w + 0.75 = -3.092329219 Simplifying w + 0.75 = -3.092329219 Reorder the terms: 0.75 + w = -3.092329219 Solving 0.75 + w = -3.092329219 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -3.092329219 + -0.75 w = -3.092329219 + -0.75 Combine like terms: -3.092329219 + -0.75 = -3.842329219 w = -3.842329219 Simplifying w = -3.842329219Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.342329219, -3.842329219}
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