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Simplifying (3x + 81)(8x + 6) = 90 Reorder the terms: (81 + 3x)(8x + 6) = 90 Reorder the terms: (81 + 3x)(6 + 8x) = 90 Multiply (81 + 3x) * (6 + 8x) (81(6 + 8x) + 3x * (6 + 8x)) = 90 ((6 * 81 + 8x * 81) + 3x * (6 + 8x)) = 90 ((486 + 648x) + 3x * (6 + 8x)) = 90 (486 + 648x + (6 * 3x + 8x * 3x)) = 90 (486 + 648x + (18x + 24x2)) = 90 Combine like terms: 648x + 18x = 666x (486 + 666x + 24x2) = 90 Solving 486 + 666x + 24x2 = 90 Solving for variable 'x'. Reorder the terms: 486 + -90 + 666x + 24x2 = 90 + -90 Combine like terms: 486 + -90 = 396 396 + 666x + 24x2 = 90 + -90 Combine like terms: 90 + -90 = 0 396 + 666x + 24x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(66 + 111x + 4x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(66 + 111x + 4x2)' equal to zero and attempt to solve: Simplifying 66 + 111x + 4x2 = 0 Solving 66 + 111x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 16.5 + 27.75x + x2 = 0 Move the constant term to the right: Add '-16.5' to each side of the equation. 16.5 + 27.75x + -16.5 + x2 = 0 + -16.5 Reorder the terms: 16.5 + -16.5 + 27.75x + x2 = 0 + -16.5 Combine like terms: 16.5 + -16.5 = 0.0 0.0 + 27.75x + x2 = 0 + -16.5 27.75x + x2 = 0 + -16.5 Combine like terms: 0 + -16.5 = -16.5 27.75x + x2 = -16.5 The x term is 27.75x. Take half its coefficient (13.875). Square it (192.515625) and add it to both sides. Add '192.515625' to each side of the equation. 27.75x + 192.515625 + x2 = -16.5 + 192.515625 Reorder the terms: 192.515625 + 27.75x + x2 = -16.5 + 192.515625 Combine like terms: -16.5 + 192.515625 = 176.015625 192.515625 + 27.75x + x2 = 176.015625 Factor a perfect square on the left side: (x + 13.875)(x + 13.875) = 176.015625 Calculate the square root of the right side: 13.267088038 Break this problem into two subproblems by setting (x + 13.875) equal to 13.267088038 and -13.267088038.Subproblem 1
x + 13.875 = 13.267088038 Simplifying x + 13.875 = 13.267088038 Reorder the terms: 13.875 + x = 13.267088038 Solving 13.875 + x = 13.267088038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.875' to each side of the equation. 13.875 + -13.875 + x = 13.267088038 + -13.875 Combine like terms: 13.875 + -13.875 = 0.000 0.000 + x = 13.267088038 + -13.875 x = 13.267088038 + -13.875 Combine like terms: 13.267088038 + -13.875 = -0.607911962 x = -0.607911962 Simplifying x = -0.607911962Subproblem 2
x + 13.875 = -13.267088038 Simplifying x + 13.875 = -13.267088038 Reorder the terms: 13.875 + x = -13.267088038 Solving 13.875 + x = -13.267088038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.875' to each side of the equation. 13.875 + -13.875 + x = -13.267088038 + -13.875 Combine like terms: 13.875 + -13.875 = 0.000 0.000 + x = -13.267088038 + -13.875 x = -13.267088038 + -13.875 Combine like terms: -13.267088038 + -13.875 = -27.142088038 x = -27.142088038 Simplifying x = -27.142088038Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.607911962, -27.142088038}Solution
x = {-0.607911962, -27.142088038}
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