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Simplifying 64(x + 5)(x + 20) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 64(5 + x)(x + 20) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 64(5 + x)(20 + x) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Multiply (5 + x) * (20 + x) 64(5(20 + x) + x(20 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64((20 * 5 + x * 5) + x(20 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64((100 + 5x) + x(20 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64(100 + 5x + (20 * x + x * x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64(100 + 5x + (20x + x2)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Combine like terms: 5x + 20x = 25x 64(100 + 25x + x2) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) (100 * 64 + 25x * 64 + x2 * 64) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) (6400 + 1600x + 64x2) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 6400 + 1600x + 64x2 + 4(12 + x)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 6400 + 1600x + 64x2 + 4(12 + x)(4 + x) = 16(x + 4)(x + 16) Multiply (12 + x) * (4 + x) 6400 + 1600x + 64x2 + 4(12(4 + x) + x(4 + x)) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + 4((4 * 12 + x * 12) + x(4 + x)) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + 4((48 + 12x) + x(4 + x)) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + 4(48 + 12x + (4 * x + x * x)) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + 4(48 + 12x + (4x + x2)) = 16(x + 4)(x + 16) Combine like terms: 12x + 4x = 16x 6400 + 1600x + 64x2 + 4(48 + 16x + x2) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + (48 * 4 + 16x * 4 + x2 * 4) = 16(x + 4)(x + 16) 6400 + 1600x + 64x2 + (192 + 64x + 4x2) = 16(x + 4)(x + 16) Reorder the terms: 6400 + 192 + 1600x + 64x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 6400 + 192 = 6592 6592 + 1600x + 64x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 1600x + 64x = 1664x 6592 + 1664x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 64x2 + 4x2 = 68x2 6592 + 1664x + 68x2 = 16(x + 4)(x + 16) Reorder the terms: 6592 + 1664x + 68x2 = 16(4 + x)(x + 16) Reorder the terms: 6592 + 1664x + 68x2 = 16(4 + x)(16 + x) Multiply (4 + x) * (16 + x) 6592 + 1664x + 68x2 = 16(4(16 + x) + x(16 + x)) 6592 + 1664x + 68x2 = 16((16 * 4 + x * 4) + x(16 + x)) 6592 + 1664x + 68x2 = 16((64 + 4x) + x(16 + x)) 6592 + 1664x + 68x2 = 16(64 + 4x + (16 * x + x * x)) 6592 + 1664x + 68x2 = 16(64 + 4x + (16x + x2)) Combine like terms: 4x + 16x = 20x 6592 + 1664x + 68x2 = 16(64 + 20x + x2) 6592 + 1664x + 68x2 = (64 * 16 + 20x * 16 + x2 * 16) 6592 + 1664x + 68x2 = (1024 + 320x + 16x2) Solving 6592 + 1664x + 68x2 = 1024 + 320x + 16x2 Solving for variable 'x'. Reorder the terms: 6592 + -1024 + 1664x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 6592 + -1024 = 5568 5568 + 1664x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 1664x + -320x = 1344x 5568 + 1344x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 68x2 + -16x2 = 52x2 5568 + 1344x + 52x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Reorder the terms: 5568 + 1344x + 52x2 = 1024 + -1024 + 320x + -320x + 16x2 + -16x2 Combine like terms: 1024 + -1024 = 0 5568 + 1344x + 52x2 = 0 + 320x + -320x + 16x2 + -16x2 5568 + 1344x + 52x2 = 320x + -320x + 16x2 + -16x2 Combine like terms: 320x + -320x = 0 5568 + 1344x + 52x2 = 0 + 16x2 + -16x2 5568 + 1344x + 52x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 5568 + 1344x + 52x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1392 + 336x + 13x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1392 + 336x + 13x2)' equal to zero and attempt to solve: Simplifying 1392 + 336x + 13x2 = 0 Solving 1392 + 336x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 107.0769231 + 25.84615385x + x2 = 0 Move the constant term to the right: Add '-107.0769231' to each side of the equation. 107.0769231 + 25.84615385x + -107.0769231 + x2 = 0 + -107.0769231 Reorder the terms: 107.0769231 + -107.0769231 + 25.84615385x + x2 = 0 + -107.0769231 Combine like terms: 107.0769231 + -107.0769231 = 0.0000000 0.0000000 + 25.84615385x + x2 = 0 + -107.0769231 25.84615385x + x2 = 0 + -107.0769231 Combine like terms: 0 + -107.0769231 = -107.0769231 25.84615385x + x2 = -107.0769231 The x term is 25.84615385x. Take half its coefficient (12.92307693). Square it (167.0059173) and add it to both sides. Add '167.0059173' to each side of the equation. 25.84615385x + 167.0059173 + x2 = -107.0769231 + 167.0059173 Reorder the terms: 167.0059173 + 25.84615385x + x2 = -107.0769231 + 167.0059173 Combine like terms: -107.0769231 + 167.0059173 = 59.9289942 167.0059173 + 25.84615385x + x2 = 59.9289942 Factor a perfect square on the left side: (x + 12.92307693)(x + 12.92307693) = 59.9289942 Calculate the square root of the right side: 7.741381931 Break this problem into two subproblems by setting (x + 12.92307693) equal to 7.741381931 and -7.741381931.Subproblem 1
x + 12.92307693 = 7.741381931 Simplifying x + 12.92307693 = 7.741381931 Reorder the terms: 12.92307693 + x = 7.741381931 Solving 12.92307693 + x = 7.741381931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.92307693' to each side of the equation. 12.92307693 + -12.92307693 + x = 7.741381931 + -12.92307693 Combine like terms: 12.92307693 + -12.92307693 = 0.00000000 0.00000000 + x = 7.741381931 + -12.92307693 x = 7.741381931 + -12.92307693 Combine like terms: 7.741381931 + -12.92307693 = -5.181694999 x = -5.181694999 Simplifying x = -5.181694999Subproblem 2
x + 12.92307693 = -7.741381931 Simplifying x + 12.92307693 = -7.741381931 Reorder the terms: 12.92307693 + x = -7.741381931 Solving 12.92307693 + x = -7.741381931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.92307693' to each side of the equation. 12.92307693 + -12.92307693 + x = -7.741381931 + -12.92307693 Combine like terms: 12.92307693 + -12.92307693 = 0.00000000 0.00000000 + x = -7.741381931 + -12.92307693 x = -7.741381931 + -12.92307693 Combine like terms: -7.741381931 + -12.92307693 = -20.664458861 x = -20.664458861 Simplifying x = -20.664458861Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.181694999, -20.664458861}Solution
x = {-5.181694999, -20.664458861}
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