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Simplifying -4w2 + -12w + -7 = 0 Reorder the terms: -7 + -12w + -4w2 = 0 Solving -7 + -12w + -4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '-1'. -1(7 + 12w + 4w2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(7 + 12w + 4w2)' equal to zero and attempt to solve: Simplifying 7 + 12w + 4w2 = 0 Solving 7 + 12w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 3w + w2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3w + -1.75 + w2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3w + w2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3w + w2 = 0 + -1.75 3w + w2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3w + w2 = -1.75 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = -1.75 + 2.25 Reorder the terms: 2.25 + 3w + w2 = -1.75 + 2.25 Combine like terms: -1.75 + 2.25 = 0.5 2.25 + 3w + w2 = 0.5 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (w + 1.5) equal to 0.707106781 and -0.707106781.Subproblem 1
w + 1.5 = 0.707106781 Simplifying w + 1.5 = 0.707106781 Reorder the terms: 1.5 + w = 0.707106781 Solving 1.5 + w = 0.707106781 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 0.707106781 + -1.5 w = 0.707106781 + -1.5 Combine like terms: 0.707106781 + -1.5 = -0.792893219 w = -0.792893219 Simplifying w = -0.792893219Subproblem 2
w + 1.5 = -0.707106781 Simplifying w + 1.5 = -0.707106781 Reorder the terms: 1.5 + w = -0.707106781 Solving 1.5 + w = -0.707106781 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -0.707106781 + -1.5 w = -0.707106781 + -1.5 Combine like terms: -0.707106781 + -1.5 = -2.207106781 w = -2.207106781 Simplifying w = -2.207106781Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.792893219, -2.207106781}Solution
w = {-0.792893219, -2.207106781}
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