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Simplifying (-7w2 + -2w + -1) + -1(-5w2 + 3w + -2) = 0 Reorder the terms: (-1 + -2w + -7w2) + -1(-5w2 + 3w + -2) = 0 Remove parenthesis around (-1 + -2w + -7w2) -1 + -2w + -7w2 + -1(-5w2 + 3w + -2) = 0 Reorder the terms: -1 + -2w + -7w2 + -1(-2 + 3w + -5w2) = 0 -1 + -2w + -7w2 + (-2 * -1 + 3w * -1 + -5w2 * -1) = 0 -1 + -2w + -7w2 + (2 + -3w + 5w2) = 0 Reorder the terms: -1 + 2 + -2w + -3w + -7w2 + 5w2 = 0 Combine like terms: -1 + 2 = 1 1 + -2w + -3w + -7w2 + 5w2 = 0 Combine like terms: -2w + -3w = -5w 1 + -5w + -7w2 + 5w2 = 0 Combine like terms: -7w2 + 5w2 = -2w2 1 + -5w + -2w2 = 0 Solving 1 + -5w + -2w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + 2.5w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 2.5w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 2.5w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 2.5w + w2 = 0 + 0.5 2.5w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 2.5w + w2 = 0.5 The w term is 2.5w. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5w + 1.5625 + w2 = 0.5 + 1.5625 Reorder the terms: 1.5625 + 2.5w + w2 = 0.5 + 1.5625 Combine like terms: 0.5 + 1.5625 = 2.0625 1.5625 + 2.5w + w2 = 2.0625 Factor a perfect square on the left side: (w + 1.25)(w + 1.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (w + 1.25) equal to 1.436140662 and -1.436140662.Subproblem 1
w + 1.25 = 1.436140662 Simplifying w + 1.25 = 1.436140662 Reorder the terms: 1.25 + w = 1.436140662 Solving 1.25 + w = 1.436140662 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = 1.436140662 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = 1.436140662 + -1.25 w = 1.436140662 + -1.25 Combine like terms: 1.436140662 + -1.25 = 0.186140662 w = 0.186140662 Simplifying w = 0.186140662Subproblem 2
w + 1.25 = -1.436140662 Simplifying w + 1.25 = -1.436140662 Reorder the terms: 1.25 + w = -1.436140662 Solving 1.25 + w = -1.436140662 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = -1.436140662 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = -1.436140662 + -1.25 w = -1.436140662 + -1.25 Combine like terms: -1.436140662 + -1.25 = -2.686140662 w = -2.686140662 Simplifying w = -2.686140662Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.186140662, -2.686140662}
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