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Simplifying (2 + -1u)(5u + 9) = 9 Reorder the terms: (2 + -1u)(9 + 5u) = 9 Multiply (2 + -1u) * (9 + 5u) (2(9 + 5u) + -1u * (9 + 5u)) = 9 ((9 * 2 + 5u * 2) + -1u * (9 + 5u)) = 9 ((18 + 10u) + -1u * (9 + 5u)) = 9 (18 + 10u + (9 * -1u + 5u * -1u)) = 9 (18 + 10u + (-9u + -5u2)) = 9 Combine like terms: 10u + -9u = 1u (18 + 1u + -5u2) = 9 Solving 18 + 1u + -5u2 = 9 Solving for variable 'u'. Reorder the terms: 18 + -9 + 1u + -5u2 = 9 + -9 Combine like terms: 18 + -9 = 9 9 + 1u + -5u2 = 9 + -9 Combine like terms: 9 + -9 = 0 9 + 1u + -5u2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1.8 + -0.2u + u2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + -0.2u + 1.8 + u2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + -0.2u + u2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + -0.2u + u2 = 0 + 1.8 -0.2u + u2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 -0.2u + u2 = 1.8 The u term is -0.2u. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2u + 0.01 + u2 = 1.8 + 0.01 Reorder the terms: 0.01 + -0.2u + u2 = 1.8 + 0.01 Combine like terms: 1.8 + 0.01 = 1.81 0.01 + -0.2u + u2 = 1.81 Factor a perfect square on the left side: (u + -0.1)(u + -0.1) = 1.81 Calculate the square root of the right side: 1.345362405 Break this problem into two subproblems by setting (u + -0.1) equal to 1.345362405 and -1.345362405.Subproblem 1
u + -0.1 = 1.345362405 Simplifying u + -0.1 = 1.345362405 Reorder the terms: -0.1 + u = 1.345362405 Solving -0.1 + u = 1.345362405 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + u = 1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + u = 1.345362405 + 0.1 u = 1.345362405 + 0.1 Combine like terms: 1.345362405 + 0.1 = 1.445362405 u = 1.445362405 Simplifying u = 1.445362405Subproblem 2
u + -0.1 = -1.345362405 Simplifying u + -0.1 = -1.345362405 Reorder the terms: -0.1 + u = -1.345362405 Solving -0.1 + u = -1.345362405 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + u = -1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + u = -1.345362405 + 0.1 u = -1.345362405 + 0.1 Combine like terms: -1.345362405 + 0.1 = -1.245362405 u = -1.245362405 Simplifying u = -1.245362405Solution
The solution to the problem is based on the solutions from the subproblems. u = {1.445362405, -1.245362405}
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