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Simplifying (4w2 + -4w + -6) + -1(2w2 + 3w + 6) = 0 Reorder the terms: (-6 + -4w + 4w2) + -1(2w2 + 3w + 6) = 0 Remove parenthesis around (-6 + -4w + 4w2) -6 + -4w + 4w2 + -1(2w2 + 3w + 6) = 0 Reorder the terms: -6 + -4w + 4w2 + -1(6 + 3w + 2w2) = 0 -6 + -4w + 4w2 + (6 * -1 + 3w * -1 + 2w2 * -1) = 0 -6 + -4w + 4w2 + (-6 + -3w + -2w2) = 0 Reorder the terms: -6 + -6 + -4w + -3w + 4w2 + -2w2 = 0 Combine like terms: -6 + -6 = -12 -12 + -4w + -3w + 4w2 + -2w2 = 0 Combine like terms: -4w + -3w = -7w -12 + -7w + 4w2 + -2w2 = 0 Combine like terms: 4w2 + -2w2 = 2w2 -12 + -7w + 2w2 = 0 Solving -12 + -7w + 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'. -6 + -3.5w + w2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -3.5w + 6 + w2 = 0 + 6 Reorder the terms: -6 + 6 + -3.5w + w2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -3.5w + w2 = 0 + 6 -3.5w + w2 = 0 + 6 Combine like terms: 0 + 6 = 6 -3.5w + w2 = 6 The w term is -3.5w. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5w + 3.0625 + w2 = 6 + 3.0625 Reorder the terms: 3.0625 + -3.5w + w2 = 6 + 3.0625 Combine like terms: 6 + 3.0625 = 9.0625 3.0625 + -3.5w + w2 = 9.0625 Factor a perfect square on the left side: (w + -1.75)(w + -1.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (w + -1.75) equal to 3.010398645 and -3.010398645.Subproblem 1
w + -1.75 = 3.010398645 Simplifying w + -1.75 = 3.010398645 Reorder the terms: -1.75 + w = 3.010398645 Solving -1.75 + w = 3.010398645 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = 3.010398645 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = 3.010398645 + 1.75 w = 3.010398645 + 1.75 Combine like terms: 3.010398645 + 1.75 = 4.760398645 w = 4.760398645 Simplifying w = 4.760398645Subproblem 2
w + -1.75 = -3.010398645 Simplifying w + -1.75 = -3.010398645 Reorder the terms: -1.75 + w = -3.010398645 Solving -1.75 + w = -3.010398645 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = -3.010398645 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = -3.010398645 + 1.75 w = -3.010398645 + 1.75 Combine like terms: -3.010398645 + 1.75 = -1.260398645 w = -1.260398645 Simplifying w = -1.260398645Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.760398645, -1.260398645}
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