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Simplifying (30 + 2x)(24 + 2x) = 232 Multiply (30 + 2x) * (24 + 2x) (30(24 + 2x) + 2x * (24 + 2x)) = 232 ((24 * 30 + 2x * 30) + 2x * (24 + 2x)) = 232 ((720 + 60x) + 2x * (24 + 2x)) = 232 (720 + 60x + (24 * 2x + 2x * 2x)) = 232 (720 + 60x + (48x + 4x2)) = 232 Combine like terms: 60x + 48x = 108x (720 + 108x + 4x2) = 232 Solving 720 + 108x + 4x2 = 232 Solving for variable 'x'. Reorder the terms: 720 + -232 + 108x + 4x2 = 232 + -232 Combine like terms: 720 + -232 = 488 488 + 108x + 4x2 = 232 + -232 Combine like terms: 232 + -232 = 0 488 + 108x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(122 + 27x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(122 + 27x + x2)' equal to zero and attempt to solve: Simplifying 122 + 27x + x2 = 0 Solving 122 + 27x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-122' to each side of the equation. 122 + 27x + -122 + x2 = 0 + -122 Reorder the terms: 122 + -122 + 27x + x2 = 0 + -122 Combine like terms: 122 + -122 = 0 0 + 27x + x2 = 0 + -122 27x + x2 = 0 + -122 Combine like terms: 0 + -122 = -122 27x + x2 = -122 The x term is 27x. Take half its coefficient (13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. 27x + 182.25 + x2 = -122 + 182.25 Reorder the terms: 182.25 + 27x + x2 = -122 + 182.25 Combine like terms: -122 + 182.25 = 60.25 182.25 + 27x + x2 = 60.25 Factor a perfect square on the left side: (x + 13.5)(x + 13.5) = 60.25 Calculate the square root of the right side: 7.762087348 Break this problem into two subproblems by setting (x + 13.5) equal to 7.762087348 and -7.762087348.Subproblem 1
x + 13.5 = 7.762087348 Simplifying x + 13.5 = 7.762087348 Reorder the terms: 13.5 + x = 7.762087348 Solving 13.5 + x = 7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + x = 7.762087348 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + x = 7.762087348 + -13.5 x = 7.762087348 + -13.5 Combine like terms: 7.762087348 + -13.5 = -5.737912652 x = -5.737912652 Simplifying x = -5.737912652Subproblem 2
x + 13.5 = -7.762087348 Simplifying x + 13.5 = -7.762087348 Reorder the terms: 13.5 + x = -7.762087348 Solving 13.5 + x = -7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + x = -7.762087348 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + x = -7.762087348 + -13.5 x = -7.762087348 + -13.5 Combine like terms: -7.762087348 + -13.5 = -21.262087348 x = -21.262087348 Simplifying x = -21.262087348Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.737912652, -21.262087348}Solution
x = {-5.737912652, -21.262087348}
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