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Simplifying 5w(-7w + 3) + 2w(-2w2 + 19w + 2) = 0 Reorder the terms: 5w(3 + -7w) + 2w(-2w2 + 19w + 2) = 0 (3 * 5w + -7w * 5w) + 2w(-2w2 + 19w + 2) = 0 (15w + -35w2) + 2w(-2w2 + 19w + 2) = 0 Reorder the terms: 15w + -35w2 + 2w(2 + 19w + -2w2) = 0 15w + -35w2 + (2 * 2w + 19w * 2w + -2w2 * 2w) = 0 15w + -35w2 + (4w + 38w2 + -4w3) = 0 Reorder the terms: 15w + 4w + -35w2 + 38w2 + -4w3 = 0 Combine like terms: 15w + 4w = 19w 19w + -35w2 + 38w2 + -4w3 = 0 Combine like terms: -35w2 + 38w2 = 3w2 19w + 3w2 + -4w3 = 0 Solving 19w + 3w2 + -4w3 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), 'w'. w(19 + 3w + -4w2) = 0Subproblem 1
Set the factor 'w' equal to zero and attempt to solve: Simplifying w = 0 Solving w = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w = 0Subproblem 2
Set the factor '(19 + 3w + -4w2)' equal to zero and attempt to solve: Simplifying 19 + 3w + -4w2 = 0 Solving 19 + 3w + -4w2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -4.75 + -0.75w + w2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + -0.75w + 4.75 + w2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + -0.75w + w2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + -0.75w + w2 = 0 + 4.75 -0.75w + w2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 -0.75w + w2 = 4.75 The w term is -0.75w. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75w + 0.140625 + w2 = 4.75 + 0.140625 Reorder the terms: 0.140625 + -0.75w + w2 = 4.75 + 0.140625 Combine like terms: 4.75 + 0.140625 = 4.890625 0.140625 + -0.75w + w2 = 4.890625 Factor a perfect square on the left side: (w + -0.375)(w + -0.375) = 4.890625 Calculate the square root of the right side: 2.211475752 Break this problem into two subproblems by setting (w + -0.375) equal to 2.211475752 and -2.211475752.Subproblem 1
w + -0.375 = 2.211475752 Simplifying w + -0.375 = 2.211475752 Reorder the terms: -0.375 + w = 2.211475752 Solving -0.375 + w = 2.211475752 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = 2.211475752 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = 2.211475752 + 0.375 w = 2.211475752 + 0.375 Combine like terms: 2.211475752 + 0.375 = 2.586475752 w = 2.586475752 Simplifying w = 2.586475752Subproblem 2
w + -0.375 = -2.211475752 Simplifying w + -0.375 = -2.211475752 Reorder the terms: -0.375 + w = -2.211475752 Solving -0.375 + w = -2.211475752 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = -2.211475752 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = -2.211475752 + 0.375 w = -2.211475752 + 0.375 Combine like terms: -2.211475752 + 0.375 = -1.836475752 w = -1.836475752 Simplifying w = -1.836475752Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.586475752, -1.836475752}Solution
w = {0, 2.586475752, -1.836475752}
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