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Simplifying 6w2 + -12w + 3 = 0 Reorder the terms: 3 + -12w + 6w2 = 0 Solving 3 + -12w + 6w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '3'. 3(1 + -4w + 2w2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + -4w + 2w2)' equal to zero and attempt to solve: Simplifying 1 + -4w + 2w2 = 0 Solving 1 + -4w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2w + w2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2w + -0.5 + w2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2w + w2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2w + w2 = 0 + -0.5 -2w + w2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2w + w2 = -0.5 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 = -0.5 + 1 Reorder the terms: 1 + -2w + w2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2w + w2 = 0.5 Factor a perfect square on the left side: (w + -1)(w + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (w + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
w + -1 = 0.707106781 Simplifying w + -1 = 0.707106781 Reorder the terms: -1 + w = 0.707106781 Solving -1 + w = 0.707106781 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 = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + w = 0.707106781 + 1 w = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 w = 1.707106781 Simplifying w = 1.707106781Subproblem 2
w + -1 = -0.707106781 Simplifying w + -1 = -0.707106781 Reorder the terms: -1 + w = -0.707106781 Solving -1 + w = -0.707106781 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 = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + w = -0.707106781 + 1 w = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 w = 0.292893219 Simplifying w = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.707106781, 0.292893219}Solution
w = {1.707106781, 0.292893219}
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