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Simplifying 12u2 + 3u = 32 Reorder the terms: 3u + 12u2 = 32 Solving 3u + 12u2 = 32 Solving for variable 'u'. Reorder the terms: -32 + 3u + 12u2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 3u + 12u2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -2.666666667 + 0.25u + u2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + 0.25u + 2.666666667 + u2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + 0.25u + u2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + 0.25u + u2 = 0 + 2.666666667 0.25u + u2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 0.25u + u2 = 2.666666667 The u term is 0.25u. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25u + 0.015625 + u2 = 2.666666667 + 0.015625 Reorder the terms: 0.015625 + 0.25u + u2 = 2.666666667 + 0.015625 Combine like terms: 2.666666667 + 0.015625 = 2.682291667 0.015625 + 0.25u + u2 = 2.682291667 Factor a perfect square on the left side: (u + 0.125)(u + 0.125) = 2.682291667 Calculate the square root of the right side: 1.637770334 Break this problem into two subproblems by setting (u + 0.125) equal to 1.637770334 and -1.637770334.Subproblem 1
u + 0.125 = 1.637770334 Simplifying u + 0.125 = 1.637770334 Reorder the terms: 0.125 + u = 1.637770334 Solving 0.125 + u = 1.637770334 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + u = 1.637770334 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + u = 1.637770334 + -0.125 u = 1.637770334 + -0.125 Combine like terms: 1.637770334 + -0.125 = 1.512770334 u = 1.512770334 Simplifying u = 1.512770334Subproblem 2
u + 0.125 = -1.637770334 Simplifying u + 0.125 = -1.637770334 Reorder the terms: 0.125 + u = -1.637770334 Solving 0.125 + u = -1.637770334 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + u = -1.637770334 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + u = -1.637770334 + -0.125 u = -1.637770334 + -0.125 Combine like terms: -1.637770334 + -0.125 = -1.762770334 u = -1.762770334 Simplifying u = -1.762770334Solution
The solution to the problem is based on the solutions from the subproblems. u = {1.512770334, -1.762770334}
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