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Simplifying (-7w2 + -3w + 4) + (-5w2 + 2) = 0 Reorder the terms: (4 + -3w + -7w2) + (-5w2 + 2) = 0 Remove parenthesis around (4 + -3w + -7w2) 4 + -3w + -7w2 + (-5w2 + 2) = 0 Reorder the terms: 4 + -3w + -7w2 + (2 + -5w2) = 0 Remove parenthesis around (2 + -5w2) 4 + -3w + -7w2 + 2 + -5w2 = 0 Reorder the terms: 4 + 2 + -3w + -7w2 + -5w2 = 0 Combine like terms: 4 + 2 = 6 6 + -3w + -7w2 + -5w2 = 0 Combine like terms: -7w2 + -5w2 = -12w2 6 + -3w + -12w2 = 0 Solving 6 + -3w + -12w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '3'. 3(2 + -1w + -4w2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(2 + -1w + -4w2)' equal to zero and attempt to solve: Simplifying 2 + -1w + -4w2 = 0 Solving 2 + -1w + -4w2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.5 + 0.25w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.25w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.25w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.25w + w2 = 0 + 0.5 0.25w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.25w + w2 = 0.5 The w term is 0.25w. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25w + 0.015625 + w2 = 0.5 + 0.015625 Reorder the terms: 0.015625 + 0.25w + w2 = 0.5 + 0.015625 Combine like terms: 0.5 + 0.015625 = 0.515625 0.015625 + 0.25w + w2 = 0.515625 Factor a perfect square on the left side: (w + 0.125)(w + 0.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (w + 0.125) equal to 0.718070331 and -0.718070331.Subproblem 1
w + 0.125 = 0.718070331 Simplifying w + 0.125 = 0.718070331 Reorder the terms: 0.125 + w = 0.718070331 Solving 0.125 + w = 0.718070331 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + w = 0.718070331 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + w = 0.718070331 + -0.125 w = 0.718070331 + -0.125 Combine like terms: 0.718070331 + -0.125 = 0.593070331 w = 0.593070331 Simplifying w = 0.593070331Subproblem 2
w + 0.125 = -0.718070331 Simplifying w + 0.125 = -0.718070331 Reorder the terms: 0.125 + w = -0.718070331 Solving 0.125 + w = -0.718070331 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + w = -0.718070331 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + w = -0.718070331 + -0.125 w = -0.718070331 + -0.125 Combine like terms: -0.718070331 + -0.125 = -0.843070331 w = -0.843070331 Simplifying w = -0.843070331Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.593070331, -0.843070331}Solution
w = {0.593070331, -0.843070331}
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