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Simplifying (4u + 1)(8 + -1u) = u Reorder the terms: (1 + 4u)(8 + -1u) = u Multiply (1 + 4u) * (8 + -1u) (1(8 + -1u) + 4u * (8 + -1u)) = u ((8 * 1 + -1u * 1) + 4u * (8 + -1u)) = u ((8 + -1u) + 4u * (8 + -1u)) = u (8 + -1u + (8 * 4u + -1u * 4u)) = u (8 + -1u + (32u + -4u2)) = u Combine like terms: -1u + 32u = 31u (8 + 31u + -4u2) = u Solving 8 + 31u + -4u2 = u Solving for variable 'u'. Reorder the terms: 8 + 31u + -1u + -4u2 = u + -1u Combine like terms: 31u + -1u = 30u 8 + 30u + -4u2 = u + -1u Combine like terms: u + -1u = 0 8 + 30u + -4u2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(4 + 15u + -2u2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + 15u + -2u2)' equal to zero and attempt to solve: Simplifying 4 + 15u + -2u2 = 0 Solving 4 + 15u + -2u2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + -7.5u + u2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -7.5u + 2 + u2 = 0 + 2 Reorder the terms: -2 + 2 + -7.5u + u2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -7.5u + u2 = 0 + 2 -7.5u + u2 = 0 + 2 Combine like terms: 0 + 2 = 2 -7.5u + u2 = 2 The u term is -7.5u. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5u + 14.0625 + u2 = 2 + 14.0625 Reorder the terms: 14.0625 + -7.5u + u2 = 2 + 14.0625 Combine like terms: 2 + 14.0625 = 16.0625 14.0625 + -7.5u + u2 = 16.0625 Factor a perfect square on the left side: (u + -3.75)(u + -3.75) = 16.0625 Calculate the square root of the right side: 4.007804885 Break this problem into two subproblems by setting (u + -3.75) equal to 4.007804885 and -4.007804885.Subproblem 1
u + -3.75 = 4.007804885 Simplifying u + -3.75 = 4.007804885 Reorder the terms: -3.75 + u = 4.007804885 Solving -3.75 + u = 4.007804885 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + u = 4.007804885 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + u = 4.007804885 + 3.75 u = 4.007804885 + 3.75 Combine like terms: 4.007804885 + 3.75 = 7.757804885 u = 7.757804885 Simplifying u = 7.757804885Subproblem 2
u + -3.75 = -4.007804885 Simplifying u + -3.75 = -4.007804885 Reorder the terms: -3.75 + u = -4.007804885 Solving -3.75 + u = -4.007804885 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + u = -4.007804885 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + u = -4.007804885 + 3.75 u = -4.007804885 + 3.75 Combine like terms: -4.007804885 + 3.75 = -0.257804885 u = -0.257804885 Simplifying u = -0.257804885Solution
The solution to the problem is based on the solutions from the subproblems. u = {7.757804885, -0.257804885}Solution
u = {7.757804885, -0.257804885}
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