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Simplifying 64(x + 8)(x + 16) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 64(8 + x)(x + 16) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 64(8 + x)(16 + x) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Multiply (8 + x) * (16 + x) 64(8(16 + x) + x(16 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64((16 * 8 + x * 8) + x(16 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64((128 + 8x) + x(16 + x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64(128 + 8x + (16 * x + x * x)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) 64(128 + 8x + (16x + x2)) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Combine like terms: 8x + 16x = 24x 64(128 + 24x + x2) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) (128 * 64 + 24x * 64 + x2 * 64) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) (8192 + 1536x + 64x2) + 4(x + 12)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 1536x + 64x2 + 4(12 + x)(x + 4) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 1536x + 64x2 + 4(12 + x)(4 + x) = 16(x + 4)(x + 16) Multiply (12 + x) * (4 + x) 8192 + 1536x + 64x2 + 4(12(4 + x) + x(4 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4((4 * 12 + x * 12) + x(4 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4((48 + 12x) + x(4 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4(48 + 12x + (4 * x + x * x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4(48 + 12x + (4x + x2)) = 16(x + 4)(x + 16) Combine like terms: 12x + 4x = 16x 8192 + 1536x + 64x2 + 4(48 + 16x + x2) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + (48 * 4 + 16x * 4 + x2 * 4) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + (192 + 64x + 4x2) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 192 + 1536x + 64x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 8192 + 192 = 8384 8384 + 1536x + 64x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 1536x + 64x = 1600x 8384 + 1600x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 64x2 + 4x2 = 68x2 8384 + 1600x + 68x2 = 16(x + 4)(x + 16) Reorder the terms: 8384 + 1600x + 68x2 = 16(4 + x)(x + 16) Reorder the terms: 8384 + 1600x + 68x2 = 16(4 + x)(16 + x) Multiply (4 + x) * (16 + x) 8384 + 1600x + 68x2 = 16(4(16 + x) + x(16 + x)) 8384 + 1600x + 68x2 = 16((16 * 4 + x * 4) + x(16 + x)) 8384 + 1600x + 68x2 = 16((64 + 4x) + x(16 + x)) 8384 + 1600x + 68x2 = 16(64 + 4x + (16 * x + x * x)) 8384 + 1600x + 68x2 = 16(64 + 4x + (16x + x2)) Combine like terms: 4x + 16x = 20x 8384 + 1600x + 68x2 = 16(64 + 20x + x2) 8384 + 1600x + 68x2 = (64 * 16 + 20x * 16 + x2 * 16) 8384 + 1600x + 68x2 = (1024 + 320x + 16x2) Solving 8384 + 1600x + 68x2 = 1024 + 320x + 16x2 Solving for variable 'x'. Reorder the terms: 8384 + -1024 + 1600x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 8384 + -1024 = 7360 7360 + 1600x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 1600x + -320x = 1280x 7360 + 1280x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 68x2 + -16x2 = 52x2 7360 + 1280x + 52x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Reorder the terms: 7360 + 1280x + 52x2 = 1024 + -1024 + 320x + -320x + 16x2 + -16x2 Combine like terms: 1024 + -1024 = 0 7360 + 1280x + 52x2 = 0 + 320x + -320x + 16x2 + -16x2 7360 + 1280x + 52x2 = 320x + -320x + 16x2 + -16x2 Combine like terms: 320x + -320x = 0 7360 + 1280x + 52x2 = 0 + 16x2 + -16x2 7360 + 1280x + 52x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 7360 + 1280x + 52x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1840 + 320x + 13x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1840 + 320x + 13x2)' equal to zero and attempt to solve: Simplifying 1840 + 320x + 13x2 = 0 Solving 1840 + 320x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 141.5384615 + 24.61538462x + x2 = 0 Move the constant term to the right: Add '-141.5384615' to each side of the equation. 141.5384615 + 24.61538462x + -141.5384615 + x2 = 0 + -141.5384615 Reorder the terms: 141.5384615 + -141.5384615 + 24.61538462x + x2 = 0 + -141.5384615 Combine like terms: 141.5384615 + -141.5384615 = 0.0000000 0.0000000 + 24.61538462x + x2 = 0 + -141.5384615 24.61538462x + x2 = 0 + -141.5384615 Combine like terms: 0 + -141.5384615 = -141.5384615 24.61538462x + x2 = -141.5384615 The x term is 24.61538462x. Take half its coefficient (12.30769231). Square it (151.4792900) and add it to both sides. Add '151.4792900' to each side of the equation. 24.61538462x + 151.4792900 + x2 = -141.5384615 + 151.4792900 Reorder the terms: 151.4792900 + 24.61538462x + x2 = -141.5384615 + 151.4792900 Combine like terms: -141.5384615 + 151.4792900 = 9.9408285 151.4792900 + 24.61538462x + x2 = 9.9408285 Factor a perfect square on the left side: (x + 12.30769231)(x + 12.30769231) = 9.9408285 Calculate the square root of the right side: 3.152907943 Break this problem into two subproblems by setting (x + 12.30769231) equal to 3.152907943 and -3.152907943.Subproblem 1
x + 12.30769231 = 3.152907943 Simplifying x + 12.30769231 = 3.152907943 Reorder the terms: 12.30769231 + x = 3.152907943 Solving 12.30769231 + x = 3.152907943 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.30769231' to each side of the equation. 12.30769231 + -12.30769231 + x = 3.152907943 + -12.30769231 Combine like terms: 12.30769231 + -12.30769231 = 0.00000000 0.00000000 + x = 3.152907943 + -12.30769231 x = 3.152907943 + -12.30769231 Combine like terms: 3.152907943 + -12.30769231 = -9.154784367 x = -9.154784367 Simplifying x = -9.154784367Subproblem 2
x + 12.30769231 = -3.152907943 Simplifying x + 12.30769231 = -3.152907943 Reorder the terms: 12.30769231 + x = -3.152907943 Solving 12.30769231 + x = -3.152907943 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.30769231' to each side of the equation. 12.30769231 + -12.30769231 + x = -3.152907943 + -12.30769231 Combine like terms: 12.30769231 + -12.30769231 = 0.00000000 0.00000000 + x = -3.152907943 + -12.30769231 x = -3.152907943 + -12.30769231 Combine like terms: -3.152907943 + -12.30769231 = -15.460600253 x = -15.460600253 Simplifying x = -15.460600253Solution
The solution to the problem is based on the solutions from the subproblems. x = {-9.154784367, -15.460600253}Solution
x = {-9.154784367, -15.460600253}
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