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Simplifying 6w2 + -9w + -65 = 0 Reorder the terms: -65 + -9w + 6w2 = 0 Solving -65 + -9w + 6w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -10.83333333 + -1.5w + w2 = 0 Move the constant term to the right: Add '10.83333333' to each side of the equation. -10.83333333 + -1.5w + 10.83333333 + w2 = 0 + 10.83333333 Reorder the terms: -10.83333333 + 10.83333333 + -1.5w + w2 = 0 + 10.83333333 Combine like terms: -10.83333333 + 10.83333333 = 0.00000000 0.00000000 + -1.5w + w2 = 0 + 10.83333333 -1.5w + w2 = 0 + 10.83333333 Combine like terms: 0 + 10.83333333 = 10.83333333 -1.5w + w2 = 10.83333333 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 = 10.83333333 + 0.5625 Reorder the terms: 0.5625 + -1.5w + w2 = 10.83333333 + 0.5625 Combine like terms: 10.83333333 + 0.5625 = 11.39583333 0.5625 + -1.5w + w2 = 11.39583333 Factor a perfect square on the left side: (w + -0.75)(w + -0.75) = 11.39583333 Calculate the square root of the right side: 3.375771516 Break this problem into two subproblems by setting (w + -0.75) equal to 3.375771516 and -3.375771516.Subproblem 1
w + -0.75 = 3.375771516 Simplifying w + -0.75 = 3.375771516 Reorder the terms: -0.75 + w = 3.375771516 Solving -0.75 + w = 3.375771516 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 = 3.375771516 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = 3.375771516 + 0.75 w = 3.375771516 + 0.75 Combine like terms: 3.375771516 + 0.75 = 4.125771516 w = 4.125771516 Simplifying w = 4.125771516Subproblem 2
w + -0.75 = -3.375771516 Simplifying w + -0.75 = -3.375771516 Reorder the terms: -0.75 + w = -3.375771516 Solving -0.75 + w = -3.375771516 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 = -3.375771516 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = -3.375771516 + 0.75 w = -3.375771516 + 0.75 Combine like terms: -3.375771516 + 0.75 = -2.625771516 w = -2.625771516 Simplifying w = -2.625771516Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.125771516, -2.625771516}
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