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Simplifying 2w2 + 3w + -196 = 0 Reorder the terms: -196 + 3w + 2w2 = 0 Solving -196 + 3w + 2w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -98 + 1.5w + w2 = 0 Move the constant term to the right: Add '98' to each side of the equation. -98 + 1.5w + 98 + w2 = 0 + 98 Reorder the terms: -98 + 98 + 1.5w + w2 = 0 + 98 Combine like terms: -98 + 98 = 0 0 + 1.5w + w2 = 0 + 98 1.5w + w2 = 0 + 98 Combine like terms: 0 + 98 = 98 1.5w + w2 = 98 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 98 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 98 + 0.5625 Combine like terms: 98 + 0.5625 = 98.5625 0.5625 + 1.5w + w2 = 98.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 98.5625 Calculate the square root of the right side: 9.927864826 Break this problem into two subproblems by setting (w + 0.75) equal to 9.927864826 and -9.927864826.Subproblem 1
w + 0.75 = 9.927864826 Simplifying w + 0.75 = 9.927864826 Reorder the terms: 0.75 + w = 9.927864826 Solving 0.75 + w = 9.927864826 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 9.927864826 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 9.927864826 + -0.75 w = 9.927864826 + -0.75 Combine like terms: 9.927864826 + -0.75 = 9.177864826 w = 9.177864826 Simplifying w = 9.177864826Subproblem 2
w + 0.75 = -9.927864826 Simplifying w + 0.75 = -9.927864826 Reorder the terms: 0.75 + w = -9.927864826 Solving 0.75 + w = -9.927864826 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -9.927864826 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -9.927864826 + -0.75 w = -9.927864826 + -0.75 Combine like terms: -9.927864826 + -0.75 = -10.677864826 w = -10.677864826 Simplifying w = -10.677864826Solution
The solution to the problem is based on the solutions from the subproblems. w = {9.177864826, -10.677864826}
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