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Simplifying (4w2 + -6w + 3) + (-5w2 + w + 7) = 0 Reorder the terms: (3 + -6w + 4w2) + (-5w2 + w + 7) = 0 Remove parenthesis around (3 + -6w + 4w2) 3 + -6w + 4w2 + (-5w2 + w + 7) = 0 Reorder the terms: 3 + -6w + 4w2 + (7 + w + -5w2) = 0 Remove parenthesis around (7 + w + -5w2) 3 + -6w + 4w2 + 7 + w + -5w2 = 0 Reorder the terms: 3 + 7 + -6w + w + 4w2 + -5w2 = 0 Combine like terms: 3 + 7 = 10 10 + -6w + w + 4w2 + -5w2 = 0 Combine like terms: -6w + w = -5w 10 + -5w + 4w2 + -5w2 = 0 Combine like terms: 4w2 + -5w2 = -1w2 10 + -5w + -1w2 = 0 Solving 10 + -5w + -1w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -10 + 5w + w2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 5w + 10 + w2 = 0 + 10 Reorder the terms: -10 + 10 + 5w + w2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 5w + w2 = 0 + 10 5w + w2 = 0 + 10 Combine like terms: 0 + 10 = 10 5w + w2 = 10 The w term is 5w. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5w + 6.25 + w2 = 10 + 6.25 Reorder the terms: 6.25 + 5w + w2 = 10 + 6.25 Combine like terms: 10 + 6.25 = 16.25 6.25 + 5w + w2 = 16.25 Factor a perfect square on the left side: (w + 2.5)(w + 2.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (w + 2.5) equal to 4.031128874 and -4.031128874.Subproblem 1
w + 2.5 = 4.031128874 Simplifying w + 2.5 = 4.031128874 Reorder the terms: 2.5 + w = 4.031128874 Solving 2.5 + w = 4.031128874 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = 4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = 4.031128874 + -2.5 w = 4.031128874 + -2.5 Combine like terms: 4.031128874 + -2.5 = 1.531128874 w = 1.531128874 Simplifying w = 1.531128874Subproblem 2
w + 2.5 = -4.031128874 Simplifying w + 2.5 = -4.031128874 Reorder the terms: 2.5 + w = -4.031128874 Solving 2.5 + w = -4.031128874 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = -4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = -4.031128874 + -2.5 w = -4.031128874 + -2.5 Combine like terms: -4.031128874 + -2.5 = -6.531128874 w = -6.531128874 Simplifying w = -6.531128874Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.531128874, -6.531128874}
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