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Simplifying 3(5u + 6) * 4(4u + -3) = 99 Reorder the terms: 3(6 + 5u) * 4(4u + -3) = 99 Reorder the terms: 3(6 + 5u) * 4(-3 + 4u) = 99 Reorder the terms for easier multiplication: 3 * 4(6 + 5u)(-3 + 4u) = 99 Multiply 3 * 4 12(6 + 5u)(-3 + 4u) = 99 Multiply (6 + 5u) * (-3 + 4u) 12(6(-3 + 4u) + 5u * (-3 + 4u)) = 99 12((-3 * 6 + 4u * 6) + 5u * (-3 + 4u)) = 99 12((-18 + 24u) + 5u * (-3 + 4u)) = 99 12(-18 + 24u + (-3 * 5u + 4u * 5u)) = 99 12(-18 + 24u + (-15u + 20u2)) = 99 Combine like terms: 24u + -15u = 9u 12(-18 + 9u + 20u2) = 99 (-18 * 12 + 9u * 12 + 20u2 * 12) = 99 (-216 + 108u + 240u2) = 99 Solving -216 + 108u + 240u2 = 99 Solving for variable 'u'. Reorder the terms: -216 + -99 + 108u + 240u2 = 99 + -99 Combine like terms: -216 + -99 = -315 -315 + 108u + 240u2 = 99 + -99 Combine like terms: 99 + -99 = 0 -315 + 108u + 240u2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-105 + 36u + 80u2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-105 + 36u + 80u2)' equal to zero and attempt to solve: Simplifying -105 + 36u + 80u2 = 0 Solving -105 + 36u + 80u2 = 0 Begin completing the square. Divide all terms by 80 the coefficient of the squared term: Divide each side by '80'. -1.3125 + 0.45u + u2 = 0.0 Move the constant term to the right: Add '1.3125' to each side of the equation. -1.3125 + 0.45u + 1.3125 + u2 = 0.0 + 1.3125 Reorder the terms: -1.3125 + 1.3125 + 0.45u + u2 = 0.0 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + 0.45u + u2 = 0.0 + 1.3125 0.45u + u2 = 0.0 + 1.3125 Combine like terms: 0.0 + 1.3125 = 1.3125 0.45u + u2 = 1.3125 The u term is 0.45u. Take half its coefficient (0.225). Square it (0.050625) and add it to both sides. Add '0.050625' to each side of the equation. 0.45u + 0.050625 + u2 = 1.3125 + 0.050625 Reorder the terms: 0.050625 + 0.45u + u2 = 1.3125 + 0.050625 Combine like terms: 1.3125 + 0.050625 = 1.363125 0.050625 + 0.45u + u2 = 1.363125 Factor a perfect square on the left side: (u + 0.225)(u + 0.225) = 1.363125 Calculate the square root of the right side: 1.167529443 Break this problem into two subproblems by setting (u + 0.225) equal to 1.167529443 and -1.167529443.Subproblem 1
u + 0.225 = 1.167529443 Simplifying u + 0.225 = 1.167529443 Reorder the terms: 0.225 + u = 1.167529443 Solving 0.225 + u = 1.167529443 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.225' to each side of the equation. 0.225 + -0.225 + u = 1.167529443 + -0.225 Combine like terms: 0.225 + -0.225 = 0.000 0.000 + u = 1.167529443 + -0.225 u = 1.167529443 + -0.225 Combine like terms: 1.167529443 + -0.225 = 0.942529443 u = 0.942529443 Simplifying u = 0.942529443Subproblem 2
u + 0.225 = -1.167529443 Simplifying u + 0.225 = -1.167529443 Reorder the terms: 0.225 + u = -1.167529443 Solving 0.225 + u = -1.167529443 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-0.225' to each side of the equation. 0.225 + -0.225 + u = -1.167529443 + -0.225 Combine like terms: 0.225 + -0.225 = 0.000 0.000 + u = -1.167529443 + -0.225 u = -1.167529443 + -0.225 Combine like terms: -1.167529443 + -0.225 = -1.392529443 u = -1.392529443 Simplifying u = -1.392529443Solution
The solution to the problem is based on the solutions from the subproblems. u = {0.942529443, -1.392529443}Solution
u = {0.942529443, -1.392529443}
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