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Simplifying 6w + 5w(w + -2) = 23 Reorder the terms: 6w + 5w(-2 + w) = 23 6w + (-2 * 5w + w * 5w) = 23 6w + (-10w + 5w2) = 23 Combine like terms: 6w + -10w = -4w -4w + 5w2 = 23 Solving -4w + 5w2 = 23 Solving for variable 'w'. Reorder the terms: -23 + -4w + 5w2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -4w + 5w2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4.6 + -0.8w + w2 = 0 Move the constant term to the right: Add '4.6' to each side of the equation. -4.6 + -0.8w + 4.6 + w2 = 0 + 4.6 Reorder the terms: -4.6 + 4.6 + -0.8w + w2 = 0 + 4.6 Combine like terms: -4.6 + 4.6 = 0.0 0.0 + -0.8w + w2 = 0 + 4.6 -0.8w + w2 = 0 + 4.6 Combine like terms: 0 + 4.6 = 4.6 -0.8w + w2 = 4.6 The w term is -0.8w. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8w + 0.16 + w2 = 4.6 + 0.16 Reorder the terms: 0.16 + -0.8w + w2 = 4.6 + 0.16 Combine like terms: 4.6 + 0.16 = 4.76 0.16 + -0.8w + w2 = 4.76 Factor a perfect square on the left side: (w + -0.4)(w + -0.4) = 4.76 Calculate the square root of the right side: 2.181742423 Break this problem into two subproblems by setting (w + -0.4) equal to 2.181742423 and -2.181742423.Subproblem 1
w + -0.4 = 2.181742423 Simplifying w + -0.4 = 2.181742423 Reorder the terms: -0.4 + w = 2.181742423 Solving -0.4 + w = 2.181742423 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + w = 2.181742423 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + w = 2.181742423 + 0.4 w = 2.181742423 + 0.4 Combine like terms: 2.181742423 + 0.4 = 2.581742423 w = 2.581742423 Simplifying w = 2.581742423Subproblem 2
w + -0.4 = -2.181742423 Simplifying w + -0.4 = -2.181742423 Reorder the terms: -0.4 + w = -2.181742423 Solving -0.4 + w = -2.181742423 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + w = -2.181742423 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + w = -2.181742423 + 0.4 w = -2.181742423 + 0.4 Combine like terms: -2.181742423 + 0.4 = -1.781742423 w = -1.781742423 Simplifying w = -1.781742423Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.581742423, -1.781742423}
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