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Simplifying (-1u2 + 3u + -7) + (5u2 + 2u + 6) = 0 Reorder the terms: (-7 + 3u + -1u2) + (5u2 + 2u + 6) = 0 Remove parenthesis around (-7 + 3u + -1u2) -7 + 3u + -1u2 + (5u2 + 2u + 6) = 0 Reorder the terms: -7 + 3u + -1u2 + (6 + 2u + 5u2) = 0 Remove parenthesis around (6 + 2u + 5u2) -7 + 3u + -1u2 + 6 + 2u + 5u2 = 0 Reorder the terms: -7 + 6 + 3u + 2u + -1u2 + 5u2 = 0 Combine like terms: -7 + 6 = -1 -1 + 3u + 2u + -1u2 + 5u2 = 0 Combine like terms: 3u + 2u = 5u -1 + 5u + -1u2 + 5u2 = 0 Combine like terms: -1u2 + 5u2 = 4u2 -1 + 5u + 4u2 = 0 Solving -1 + 5u + 4u2 = 0 Solving for variable 'u'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 1.25u + u2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 1.25u + 0.25 + u2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 1.25u + u2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 1.25u + u2 = 0 + 0.25 1.25u + u2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 1.25u + u2 = 0.25 The u term is 1.25u. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25u + 0.390625 + u2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + 1.25u + u2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + 1.25u + u2 = 0.640625 Factor a perfect square on the left side: (u + 0.625)(u + 0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (u + 0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
u + 0.625 = 0.80039053 Simplifying u + 0.625 = 0.80039053 Reorder the terms: 0.625 + u = 0.80039053 Solving 0.625 + u = 0.80039053 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + u = 0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + u = 0.80039053 + -0.625 u = 0.80039053 + -0.625 Combine like terms: 0.80039053 + -0.625 = 0.17539053 u = 0.17539053 Simplifying u = 0.17539053Subproblem 2
u + 0.625 = -0.80039053 Simplifying u + 0.625 = -0.80039053 Reorder the terms: 0.625 + u = -0.80039053 Solving 0.625 + u = -0.80039053 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + u = -0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + u = -0.80039053 + -0.625 u = -0.80039053 + -0.625 Combine like terms: -0.80039053 + -0.625 = -1.42539053 u = -1.42539053 Simplifying u = -1.42539053Solution
The solution to the problem is based on the solutions from the subproblems. u = {0.17539053, -1.42539053}
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