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Simplifying 10w2 + 15w + 3 = 0 Reorder the terms: 3 + 15w + 10w2 = 0 Solving 3 + 15w + 10w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.3 + 1.5w + w2 = 0 Move the constant term to the right: Add '-0.3' to each side of the equation. 0.3 + 1.5w + -0.3 + w2 = 0 + -0.3 Reorder the terms: 0.3 + -0.3 + 1.5w + w2 = 0 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + 1.5w + w2 = 0 + -0.3 1.5w + w2 = 0 + -0.3 Combine like terms: 0 + -0.3 = -0.3 1.5w + w2 = -0.3 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 = -0.3 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = -0.3 + 0.5625 Combine like terms: -0.3 + 0.5625 = 0.2625 0.5625 + 1.5w + w2 = 0.2625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 0.2625 Calculate the square root of the right side: 0.512347538 Break this problem into two subproblems by setting (w + 0.75) equal to 0.512347538 and -0.512347538.Subproblem 1
w + 0.75 = 0.512347538 Simplifying w + 0.75 = 0.512347538 Reorder the terms: 0.75 + w = 0.512347538 Solving 0.75 + w = 0.512347538 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 = 0.512347538 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 0.512347538 + -0.75 w = 0.512347538 + -0.75 Combine like terms: 0.512347538 + -0.75 = -0.237652462 w = -0.237652462 Simplifying w = -0.237652462Subproblem 2
w + 0.75 = -0.512347538 Simplifying w + 0.75 = -0.512347538 Reorder the terms: 0.75 + w = -0.512347538 Solving 0.75 + w = -0.512347538 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 = -0.512347538 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -0.512347538 + -0.75 w = -0.512347538 + -0.75 Combine like terms: -0.512347538 + -0.75 = -1.262347538 w = -1.262347538 Simplifying w = -1.262347538Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.237652462, -1.262347538}
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