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Simplifying 8w + -4w2 + 14 = 0 Reorder the terms: 14 + 8w + -4w2 = 0 Solving 14 + 8w + -4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + 4w + -2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + 4w + -2w2)' equal to zero and attempt to solve: Simplifying 7 + 4w + -2w2 = 0 Solving 7 + 4w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3.5 + -2w + w2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -2w + 3.5 + w2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -2w + w2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -2w + w2 = 0 + 3.5 -2w + w2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -2w + w2 = 3.5 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 = 3.5 + 1 Reorder the terms: 1 + -2w + w2 = 3.5 + 1 Combine like terms: 3.5 + 1 = 4.5 1 + -2w + w2 = 4.5 Factor a perfect square on the left side: (w + -1)(w + -1) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (w + -1) equal to 2.121320344 and -2.121320344.Subproblem 1
w + -1 = 2.121320344 Simplifying w + -1 = 2.121320344 Reorder the terms: -1 + w = 2.121320344 Solving -1 + w = 2.121320344 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 = 2.121320344 + 1 Combine like terms: -1 + 1 = 0 0 + w = 2.121320344 + 1 w = 2.121320344 + 1 Combine like terms: 2.121320344 + 1 = 3.121320344 w = 3.121320344 Simplifying w = 3.121320344Subproblem 2
w + -1 = -2.121320344 Simplifying w + -1 = -2.121320344 Reorder the terms: -1 + w = -2.121320344 Solving -1 + w = -2.121320344 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 = -2.121320344 + 1 Combine like terms: -1 + 1 = 0 0 + w = -2.121320344 + 1 w = -2.121320344 + 1 Combine like terms: -2.121320344 + 1 = -1.121320344 w = -1.121320344 Simplifying w = -1.121320344Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.121320344, -1.121320344}Solution
w = {3.121320344, -1.121320344}
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