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Simplifying 4w2 + 7w = 194 Reorder the terms: 7w + 4w2 = 194 Solving 7w + 4w2 = 194 Solving for variable 'w'. Reorder the terms: -194 + 7w + 4w2 = 194 + -194 Combine like terms: 194 + -194 = 0 -194 + 7w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -48.5 + 1.75w + w2 = 0 Move the constant term to the right: Add '48.5' to each side of the equation. -48.5 + 1.75w + 48.5 + w2 = 0 + 48.5 Reorder the terms: -48.5 + 48.5 + 1.75w + w2 = 0 + 48.5 Combine like terms: -48.5 + 48.5 = 0.0 0.0 + 1.75w + w2 = 0 + 48.5 1.75w + w2 = 0 + 48.5 Combine like terms: 0 + 48.5 = 48.5 1.75w + w2 = 48.5 The w term is 1.75w. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75w + 0.765625 + w2 = 48.5 + 0.765625 Reorder the terms: 0.765625 + 1.75w + w2 = 48.5 + 0.765625 Combine like terms: 48.5 + 0.765625 = 49.265625 0.765625 + 1.75w + w2 = 49.265625 Factor a perfect square on the left side: (w + 0.875)(w + 0.875) = 49.265625 Calculate the square root of the right side: 7.018947571 Break this problem into two subproblems by setting (w + 0.875) equal to 7.018947571 and -7.018947571.Subproblem 1
w + 0.875 = 7.018947571 Simplifying w + 0.875 = 7.018947571 Reorder the terms: 0.875 + w = 7.018947571 Solving 0.875 + w = 7.018947571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = 7.018947571 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = 7.018947571 + -0.875 w = 7.018947571 + -0.875 Combine like terms: 7.018947571 + -0.875 = 6.143947571 w = 6.143947571 Simplifying w = 6.143947571Subproblem 2
w + 0.875 = -7.018947571 Simplifying w + 0.875 = -7.018947571 Reorder the terms: 0.875 + w = -7.018947571 Solving 0.875 + w = -7.018947571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = -7.018947571 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = -7.018947571 + -0.875 w = -7.018947571 + -0.875 Combine like terms: -7.018947571 + -0.875 = -7.893947571 w = -7.893947571 Simplifying w = -7.893947571Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.143947571, -7.893947571}
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