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Simplifying 41w2 + w = 30 Reorder the terms: w + 41w2 = 30 Solving w + 41w2 = 30 Solving for variable 'w'. Reorder the terms: -30 + w + 41w2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + w + 41w2 = 0 Begin completing the square. Divide all terms by 41 the coefficient of the squared term: Divide each side by '41'. -0.7317073171 + 0.02439024390w + w2 = 0 Move the constant term to the right: Add '0.7317073171' to each side of the equation. -0.7317073171 + 0.02439024390w + 0.7317073171 + w2 = 0 + 0.7317073171 Reorder the terms: -0.7317073171 + 0.7317073171 + 0.02439024390w + w2 = 0 + 0.7317073171 Combine like terms: -0.7317073171 + 0.7317073171 = 0.0000000000 0.0000000000 + 0.02439024390w + w2 = 0 + 0.7317073171 0.02439024390w + w2 = 0 + 0.7317073171 Combine like terms: 0 + 0.7317073171 = 0.7317073171 0.02439024390w + w2 = 0.7317073171 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.02439024390w + 0.25 + w2 = 0.7317073171 + 0.25 Reorder the terms: 0.25 + 0.02439024390w + w2 = 0.7317073171 + 0.25 Combine like terms: 0.7317073171 + 0.25 = 0.9817073171 0.25 + 0.02439024390w + w2 = 0.9817073171 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 0.9817073171 Calculate the square root of the right side: 0.990811444 Break this problem into two subproblems by setting (w + 0.5) equal to 0.990811444 and -0.990811444.Subproblem 1
w + 0.5 = 0.990811444 Simplifying w + 0.5 = 0.990811444 Reorder the terms: 0.5 + w = 0.990811444 Solving 0.5 + w = 0.990811444 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 0.990811444 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.990811444 + -0.5 w = 0.990811444 + -0.5 Combine like terms: 0.990811444 + -0.5 = 0.490811444 w = 0.490811444 Simplifying w = 0.490811444Subproblem 2
w + 0.5 = -0.990811444 Simplifying w + 0.5 = -0.990811444 Reorder the terms: 0.5 + w = -0.990811444 Solving 0.5 + w = -0.990811444 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -0.990811444 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -0.990811444 + -0.5 w = -0.990811444 + -0.5 Combine like terms: -0.990811444 + -0.5 = -1.490811444 w = -1.490811444 Simplifying w = -1.490811444Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.490811444, -1.490811444}
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