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Simplifying (4u2 + 1) + -1(2u2 + -3u + 5) = 0 Reorder the terms: (1 + 4u2) + -1(2u2 + -3u + 5) = 0 Remove parenthesis around (1 + 4u2) 1 + 4u2 + -1(2u2 + -3u + 5) = 0 Reorder the terms: 1 + 4u2 + -1(5 + -3u + 2u2) = 0 1 + 4u2 + (5 * -1 + -3u * -1 + 2u2 * -1) = 0 1 + 4u2 + (-5 + 3u + -2u2) = 0 Reorder the terms: 1 + -5 + 3u + 4u2 + -2u2 = 0 Combine like terms: 1 + -5 = -4 -4 + 3u + 4u2 + -2u2 = 0 Combine like terms: 4u2 + -2u2 = 2u2 -4 + 3u + 2u2 = 0 Solving -4 + 3u + 2u2 = 0 Solving for variable 'u'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 1.5u + u2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.5u + 2 + u2 = 0 + 2 Reorder the terms: -2 + 2 + 1.5u + u2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.5u + u2 = 0 + 2 1.5u + u2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.5u + u2 = 2 The u term is 1.5u. 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. 1.5u + 0.5625 + u2 = 2 + 0.5625 Reorder the terms: 0.5625 + 1.5u + u2 = 2 + 0.5625 Combine like terms: 2 + 0.5625 = 2.5625 0.5625 + 1.5u + u2 = 2.5625 Factor a perfect square on the left side: (u + 0.75)(u + 0.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (u + 0.75) equal to 1.600781059 and -1.600781059.Subproblem 1
u + 0.75 = 1.600781059 Simplifying u + 0.75 = 1.600781059 Reorder the terms: 0.75 + u = 1.600781059 Solving 0.75 + u = 1.600781059 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + u = 1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + u = 1.600781059 + -0.75 u = 1.600781059 + -0.75 Combine like terms: 1.600781059 + -0.75 = 0.850781059 u = 0.850781059 Simplifying u = 0.850781059Subproblem 2
u + 0.75 = -1.600781059 Simplifying u + 0.75 = -1.600781059 Reorder the terms: 0.75 + u = -1.600781059 Solving 0.75 + u = -1.600781059 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + u = -1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + u = -1.600781059 + -0.75 u = -1.600781059 + -0.75 Combine like terms: -1.600781059 + -0.75 = -2.350781059 u = -2.350781059 Simplifying u = -2.350781059Solution
The solution to the problem is based on the solutions from the subproblems. u = {0.850781059, -2.350781059}
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