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Simplifying 6w2 + -12w + -14 = 0 Reorder the terms: -14 + -12w + 6w2 = 0 Solving -14 + -12w + 6w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -6w + 3w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -6w + 3w2)' equal to zero and attempt to solve: Simplifying -7 + -6w + 3w2 = 0 Solving -7 + -6w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.333333333 + -2w + w2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -2w + 2.333333333 + w2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -2w + w2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -2w + w2 = 0 + 2.333333333 -2w + w2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -2w + w2 = 2.333333333 The w term is -2w. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2w + 1 + w2 = 2.333333333 + 1 Reorder the terms: 1 + -2w + w2 = 2.333333333 + 1 Combine like terms: 2.333333333 + 1 = 3.333333333 1 + -2w + w2 = 3.333333333 Factor a perfect square on the left side: (w + -1)(w + -1) = 3.333333333 Calculate the square root of the right side: 1.825741858 Break this problem into two subproblems by setting (w + -1) equal to 1.825741858 and -1.825741858.Subproblem 1
w + -1 = 1.825741858 Simplifying w + -1 = 1.825741858 Reorder the terms: -1 + w = 1.825741858 Solving -1 + w = 1.825741858 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + w = 1.825741858 + 1 Combine like terms: -1 + 1 = 0 0 + w = 1.825741858 + 1 w = 1.825741858 + 1 Combine like terms: 1.825741858 + 1 = 2.825741858 w = 2.825741858 Simplifying w = 2.825741858Subproblem 2
w + -1 = -1.825741858 Simplifying w + -1 = -1.825741858 Reorder the terms: -1 + w = -1.825741858 Solving -1 + w = -1.825741858 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + w = -1.825741858 + 1 Combine like terms: -1 + 1 = 0 0 + w = -1.825741858 + 1 w = -1.825741858 + 1 Combine like terms: -1.825741858 + 1 = -0.825741858 w = -0.825741858 Simplifying w = -0.825741858Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.825741858, -0.825741858}Solution
w = {2.825741858, -0.825741858}
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