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Simplifying 2u2 + -4u + -5 = (u + -4) Reorder the terms: -5 + -4u + 2u2 = (u + -4) Reorder the terms: -5 + -4u + 2u2 = (-4 + u) Remove parenthesis around (-4 + u) -5 + -4u + 2u2 = -4 + u Solving -5 + -4u + 2u2 = -4 + u Solving for variable 'u'. Reorder the terms: -5 + 4 + -4u + -1u + 2u2 = -4 + u + 4 + -1u Combine like terms: -5 + 4 = -1 -1 + -4u + -1u + 2u2 = -4 + u + 4 + -1u Combine like terms: -4u + -1u = -5u -1 + -5u + 2u2 = -4 + u + 4 + -1u Reorder the terms: -1 + -5u + 2u2 = -4 + 4 + u + -1u Combine like terms: -4 + 4 = 0 -1 + -5u + 2u2 = 0 + u + -1u -1 + -5u + 2u2 = u + -1u Combine like terms: u + -1u = 0 -1 + -5u + 2u2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -2.5u + u2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.5u + 0.5 + u2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.5u + u2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.5u + u2 = 0 + 0.5 -2.5u + u2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.5u + u2 = 0.5 The u term is -2.5u. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5u + 1.5625 + u2 = 0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5u + u2 = 0.5 + 1.5625 Combine like terms: 0.5 + 1.5625 = 2.0625 1.5625 + -2.5u + u2 = 2.0625 Factor a perfect square on the left side: (u + -1.25)(u + -1.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (u + -1.25) equal to 1.436140662 and -1.436140662.Subproblem 1
u + -1.25 = 1.436140662 Simplifying u + -1.25 = 1.436140662 Reorder the terms: -1.25 + u = 1.436140662 Solving -1.25 + u = 1.436140662 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + u = 1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + u = 1.436140662 + 1.25 u = 1.436140662 + 1.25 Combine like terms: 1.436140662 + 1.25 = 2.686140662 u = 2.686140662 Simplifying u = 2.686140662Subproblem 2
u + -1.25 = -1.436140662 Simplifying u + -1.25 = -1.436140662 Reorder the terms: -1.25 + u = -1.436140662 Solving -1.25 + u = -1.436140662 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + u = -1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + u = -1.436140662 + 1.25 u = -1.436140662 + 1.25 Combine like terms: -1.436140662 + 1.25 = -0.186140662 u = -0.186140662 Simplifying u = -0.186140662Solution
The solution to the problem is based on the solutions from the subproblems. u = {2.686140662, -0.186140662}
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