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Simplifying 2w2 + w = 12 Reorder the terms: w + 2w2 = 12 Solving w + 2w2 = 12 Solving for variable 'w'. Reorder the terms: -12 + w + 2w2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + 0.5w + w2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 0.5w + 6 + w2 = 0 + 6 Reorder the terms: -6 + 6 + 0.5w + w2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 0.5w + w2 = 0 + 6 0.5w + w2 = 0 + 6 Combine like terms: 0 + 6 = 6 0.5w + w2 = 6 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. 0.5w + 0.25 + w2 = 6 + 0.25 Reorder the terms: 0.25 + 0.5w + w2 = 6 + 0.25 Combine like terms: 6 + 0.25 = 6.25 0.25 + 0.5w + w2 = 6.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 6.25 Calculate the square root of the right side: 2.5 Break this problem into two subproblems by setting (w + 0.5) equal to 2.5 and -2.5.Subproblem 1
w + 0.5 = 2.5 Simplifying w + 0.5 = 2.5 Reorder the terms: 0.5 + w = 2.5 Solving 0.5 + w = 2.5 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 = 2.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.5 + -0.5 w = 2.5 + -0.5 Combine like terms: 2.5 + -0.5 = 2 w = 2 Simplifying w = 2Subproblem 2
w + 0.5 = -2.5 Simplifying w + 0.5 = -2.5 Reorder the terms: 0.5 + w = -2.5 Solving 0.5 + w = -2.5 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 = -2.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.5 + -0.5 w = -2.5 + -0.5 Combine like terms: -2.5 + -0.5 = -3 w = -3 Simplifying w = -3Solution
The solution to the problem is based on the solutions from the subproblems. w = {2, -3}
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