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Simplifying 3(x + 6)(x + 12) = 75 Reorder the terms: 3(6 + x)(x + 12) = 75 Reorder the terms: 3(6 + x)(12 + x) = 75 Multiply (6 + x) * (12 + x) 3(6(12 + x) + x(12 + x)) = 75 3((12 * 6 + x * 6) + x(12 + x)) = 75 3((72 + 6x) + x(12 + x)) = 75 3(72 + 6x + (12 * x + x * x)) = 75 3(72 + 6x + (12x + x2)) = 75 Combine like terms: 6x + 12x = 18x 3(72 + 18x + x2) = 75 (72 * 3 + 18x * 3 + x2 * 3) = 75 (216 + 54x + 3x2) = 75 Solving 216 + 54x + 3x2 = 75 Solving for variable 'x'. Reorder the terms: 216 + -75 + 54x + 3x2 = 75 + -75 Combine like terms: 216 + -75 = 141 141 + 54x + 3x2 = 75 + -75 Combine like terms: 75 + -75 = 0 141 + 54x + 3x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(47 + 18x + x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(47 + 18x + x2)' equal to zero and attempt to solve: Simplifying 47 + 18x + x2 = 0 Solving 47 + 18x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-47' to each side of the equation. 47 + 18x + -47 + x2 = 0 + -47 Reorder the terms: 47 + -47 + 18x + x2 = 0 + -47 Combine like terms: 47 + -47 = 0 0 + 18x + x2 = 0 + -47 18x + x2 = 0 + -47 Combine like terms: 0 + -47 = -47 18x + x2 = -47 The x term is 18x. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18x + 81 + x2 = -47 + 81 Reorder the terms: 81 + 18x + x2 = -47 + 81 Combine like terms: -47 + 81 = 34 81 + 18x + x2 = 34 Factor a perfect square on the left side: (x + 9)(x + 9) = 34 Calculate the square root of the right side: 5.830951895 Break this problem into two subproblems by setting (x + 9) equal to 5.830951895 and -5.830951895.Subproblem 1
x + 9 = 5.830951895 Simplifying x + 9 = 5.830951895 Reorder the terms: 9 + x = 5.830951895 Solving 9 + x = 5.830951895 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + x = 5.830951895 + -9 Combine like terms: 9 + -9 = 0 0 + x = 5.830951895 + -9 x = 5.830951895 + -9 Combine like terms: 5.830951895 + -9 = -3.169048105 x = -3.169048105 Simplifying x = -3.169048105Subproblem 2
x + 9 = -5.830951895 Simplifying x + 9 = -5.830951895 Reorder the terms: 9 + x = -5.830951895 Solving 9 + x = -5.830951895 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + x = -5.830951895 + -9 Combine like terms: 9 + -9 = 0 0 + x = -5.830951895 + -9 x = -5.830951895 + -9 Combine like terms: -5.830951895 + -9 = -14.830951895 x = -14.830951895 Simplifying x = -14.830951895Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.169048105, -14.830951895}Solution
x = {-3.169048105, -14.830951895}
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