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Simplifying 784 = 2w2 + w Reorder the terms: 784 = w + 2w2 Solving 784 = w + 2w2 Solving for variable 'w'. Reorder the terms: 784 + -1w + -2w2 = w + -1w + 2w2 + -2w2 Combine like terms: w + -1w = 0 784 + -1w + -2w2 = 0 + 2w2 + -2w2 784 + -1w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 784 + -1w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -392 + 0.5w + w2 = 0 Move the constant term to the right: Add '392' to each side of the equation. -392 + 0.5w + 392 + w2 = 0 + 392 Reorder the terms: -392 + 392 + 0.5w + w2 = 0 + 392 Combine like terms: -392 + 392 = 0 0 + 0.5w + w2 = 0 + 392 0.5w + w2 = 0 + 392 Combine like terms: 0 + 392 = 392 0.5w + w2 = 392 The w term is 0.5w. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5w + 0.0625 + w2 = 392 + 0.0625 Reorder the terms: 0.0625 + 0.5w + w2 = 392 + 0.0625 Combine like terms: 392 + 0.0625 = 392.0625 0.0625 + 0.5w + w2 = 392.0625 Factor a perfect square on the left side: (w + 0.25)(w + 0.25) = 392.0625 Calculate the square root of the right side: 19.800568174 Break this problem into two subproblems by setting (w + 0.25) equal to 19.800568174 and -19.800568174.Subproblem 1
w + 0.25 = 19.800568174 Simplifying w + 0.25 = 19.800568174 Reorder the terms: 0.25 + w = 19.800568174 Solving 0.25 + w = 19.800568174 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 19.800568174 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 19.800568174 + -0.25 w = 19.800568174 + -0.25 Combine like terms: 19.800568174 + -0.25 = 19.550568174 w = 19.550568174 Simplifying w = 19.550568174Subproblem 2
w + 0.25 = -19.800568174 Simplifying w + 0.25 = -19.800568174 Reorder the terms: 0.25 + w = -19.800568174 Solving 0.25 + w = -19.800568174 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -19.800568174 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -19.800568174 + -0.25 w = -19.800568174 + -0.25 Combine like terms: -19.800568174 + -0.25 = -20.050568174 w = -20.050568174 Simplifying w = -20.050568174Solution
The solution to the problem is based on the solutions from the subproblems. w = {19.550568174, -20.050568174}
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