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Simplifying 2w2 + 3w + -96 = 0 Reorder the terms: -96 + 3w + 2w2 = 0 Solving -96 + 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'. -48 + 1.5w + w2 = 0 Move the constant term to the right: Add '48' to each side of the equation. -48 + 1.5w + 48 + w2 = 0 + 48 Reorder the terms: -48 + 48 + 1.5w + w2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 1.5w + w2 = 0 + 48 1.5w + w2 = 0 + 48 Combine like terms: 0 + 48 = 48 1.5w + w2 = 48 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 = 48 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 48 + 0.5625 Combine like terms: 48 + 0.5625 = 48.5625 0.5625 + 1.5w + w2 = 48.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 48.5625 Calculate the square root of the right side: 6.968679932 Break this problem into two subproblems by setting (w + 0.75) equal to 6.968679932 and -6.968679932.Subproblem 1
w + 0.75 = 6.968679932 Simplifying w + 0.75 = 6.968679932 Reorder the terms: 0.75 + w = 6.968679932 Solving 0.75 + w = 6.968679932 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 = 6.968679932 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 6.968679932 + -0.75 w = 6.968679932 + -0.75 Combine like terms: 6.968679932 + -0.75 = 6.218679932 w = 6.218679932 Simplifying w = 6.218679932Subproblem 2
w + 0.75 = -6.968679932 Simplifying w + 0.75 = -6.968679932 Reorder the terms: 0.75 + w = -6.968679932 Solving 0.75 + w = -6.968679932 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 = -6.968679932 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -6.968679932 + -0.75 w = -6.968679932 + -0.75 Combine like terms: -6.968679932 + -0.75 = -7.718679932 w = -7.718679932 Simplifying w = -7.718679932Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.218679932, -7.718679932}
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