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Simplifying (7u2 + 7u + 5) + (-6u2 + -7u + -4) + -1(9u2 + -2u + -7) = 0 Reorder the terms: (5 + 7u + 7u2) + (-6u2 + -7u + -4) + -1(9u2 + -2u + -7) = 0 Remove parenthesis around (5 + 7u + 7u2) 5 + 7u + 7u2 + (-6u2 + -7u + -4) + -1(9u2 + -2u + -7) = 0 Reorder the terms: 5 + 7u + 7u2 + (-4 + -7u + -6u2) + -1(9u2 + -2u + -7) = 0 Remove parenthesis around (-4 + -7u + -6u2) 5 + 7u + 7u2 + -4 + -7u + -6u2 + -1(9u2 + -2u + -7) = 0 Reorder the terms: 5 + 7u + 7u2 + -4 + -7u + -6u2 + -1(-7 + -2u + 9u2) = 0 5 + 7u + 7u2 + -4 + -7u + -6u2 + (-7 * -1 + -2u * -1 + 9u2 * -1) = 0 5 + 7u + 7u2 + -4 + -7u + -6u2 + (7 + 2u + -9u2) = 0 Reorder the terms: 5 + -4 + 7 + 7u + -7u + 2u + 7u2 + -6u2 + -9u2 = 0 Combine like terms: 5 + -4 = 1 1 + 7 + 7u + -7u + 2u + 7u2 + -6u2 + -9u2 = 0 Combine like terms: 1 + 7 = 8 8 + 7u + -7u + 2u + 7u2 + -6u2 + -9u2 = 0 Combine like terms: 7u + -7u = 0 8 + 0 + 2u + 7u2 + -6u2 + -9u2 = 0 8 + 2u + 7u2 + -6u2 + -9u2 = 0 Combine like terms: 7u2 + -6u2 = 1u2 8 + 2u + 1u2 + -9u2 = 0 Combine like terms: 1u2 + -9u2 = -8u2 8 + 2u + -8u2 = 0 Solving 8 + 2u + -8u2 = 0 Solving for variable 'u'. Factor out the Greatest Common Factor (GCF), '2'. 2(4 + u + -4u2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + u + -4u2)' equal to zero and attempt to solve: Simplifying 4 + u + -4u2 = 0 Solving 4 + u + -4u2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + -0.25u + u2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.25u + 1 + u2 = 0 + 1 Reorder the terms: -1 + 1 + -0.25u + u2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.25u + u2 = 0 + 1 -0.25u + u2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.25u + u2 = 1 The u term is u. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.25u + 0.25 + u2 = 1 + 0.25 Reorder the terms: 0.25 + -0.25u + u2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -0.25u + u2 = 1.25 Factor a perfect square on the left side: (u + 0.5)(u + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (u + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
u + 0.5 = 1.118033989 Simplifying u + 0.5 = 1.118033989 Reorder the terms: 0.5 + u = 1.118033989 Solving 0.5 + u = 1.118033989 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + u = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + u = 1.118033989 + -0.5 u = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 u = 0.618033989 Simplifying u = 0.618033989Subproblem 2
u + 0.5 = -1.118033989 Simplifying u + 0.5 = -1.118033989 Reorder the terms: 0.5 + u = -1.118033989 Solving 0.5 + u = -1.118033989 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + u = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + u = -1.118033989 + -0.5 u = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 u = -1.618033989 Simplifying u = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. u = {0.618033989, -1.618033989}Solution
u = {0.618033989, -1.618033989}
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