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Simplifying 64(x + 8)(x + 16) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) Reorder the terms: 64(8 + x)(x + 16) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) Reorder the terms: 64(8 + x)(16 + x) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) Multiply (8 + x) * (16 + x) 64(8(16 + x) + x(16 + x)) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) 64((16 * 8 + x * 8) + x(16 + x)) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) 64((128 + 8x) + x(16 + x)) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) 64(128 + 8x + (16 * x + x * x)) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) 64(128 + 8x + (16x + x2)) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) Combine like terms: 8x + 16x = 24x 64(128 + 24x + x2) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) (128 * 64 + 24x * 64 + x2 * 64) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) (8192 + 1536x + 64x2) + 4(x + 12)(x + 8) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 1536x + 64x2 + 4(12 + x)(x + 8) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 1536x + 64x2 + 4(12 + x)(8 + x) = 16(x + 4)(x + 16) Multiply (12 + x) * (8 + x) 8192 + 1536x + 64x2 + 4(12(8 + x) + x(8 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4((8 * 12 + x * 12) + x(8 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4((96 + 12x) + x(8 + x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4(96 + 12x + (8 * x + x * x)) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + 4(96 + 12x + (8x + x2)) = 16(x + 4)(x + 16) Combine like terms: 12x + 8x = 20x 8192 + 1536x + 64x2 + 4(96 + 20x + x2) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + (96 * 4 + 20x * 4 + x2 * 4) = 16(x + 4)(x + 16) 8192 + 1536x + 64x2 + (384 + 80x + 4x2) = 16(x + 4)(x + 16) Reorder the terms: 8192 + 384 + 1536x + 80x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 8192 + 384 = 8576 8576 + 1536x + 80x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 1536x + 80x = 1616x 8576 + 1616x + 64x2 + 4x2 = 16(x + 4)(x + 16) Combine like terms: 64x2 + 4x2 = 68x2 8576 + 1616x + 68x2 = 16(x + 4)(x + 16) Reorder the terms: 8576 + 1616x + 68x2 = 16(4 + x)(x + 16) Reorder the terms: 8576 + 1616x + 68x2 = 16(4 + x)(16 + x) Multiply (4 + x) * (16 + x) 8576 + 1616x + 68x2 = 16(4(16 + x) + x(16 + x)) 8576 + 1616x + 68x2 = 16((16 * 4 + x * 4) + x(16 + x)) 8576 + 1616x + 68x2 = 16((64 + 4x) + x(16 + x)) 8576 + 1616x + 68x2 = 16(64 + 4x + (16 * x + x * x)) 8576 + 1616x + 68x2 = 16(64 + 4x + (16x + x2)) Combine like terms: 4x + 16x = 20x 8576 + 1616x + 68x2 = 16(64 + 20x + x2) 8576 + 1616x + 68x2 = (64 * 16 + 20x * 16 + x2 * 16) 8576 + 1616x + 68x2 = (1024 + 320x + 16x2) Solving 8576 + 1616x + 68x2 = 1024 + 320x + 16x2 Solving for variable 'x'. Reorder the terms: 8576 + -1024 + 1616x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 8576 + -1024 = 7552 7552 + 1616x + -320x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 1616x + -320x = 1296x 7552 + 1296x + 68x2 + -16x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Combine like terms: 68x2 + -16x2 = 52x2 7552 + 1296x + 52x2 = 1024 + 320x + 16x2 + -1024 + -320x + -16x2 Reorder the terms: 7552 + 1296x + 52x2 = 1024 + -1024 + 320x + -320x + 16x2 + -16x2 Combine like terms: 1024 + -1024 = 0 7552 + 1296x + 52x2 = 0 + 320x + -320x + 16x2 + -16x2 7552 + 1296x + 52x2 = 320x + -320x + 16x2 + -16x2 Combine like terms: 320x + -320x = 0 7552 + 1296x + 52x2 = 0 + 16x2 + -16x2 7552 + 1296x + 52x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 7552 + 1296x + 52x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1888 + 324x + 13x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1888 + 324x + 13x2)' equal to zero and attempt to solve: Simplifying 1888 + 324x + 13x2 = 0 Solving 1888 + 324x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 145.2307692 + 24.92307692x + x2 = 0 Move the constant term to the right: Add '-145.2307692' to each side of the equation. 145.2307692 + 24.92307692x + -145.2307692 + x2 = 0 + -145.2307692 Reorder the terms: 145.2307692 + -145.2307692 + 24.92307692x + x2 = 0 + -145.2307692 Combine like terms: 145.2307692 + -145.2307692 = 0.0000000 0.0000000 + 24.92307692x + x2 = 0 + -145.2307692 24.92307692x + x2 = 0 + -145.2307692 Combine like terms: 0 + -145.2307692 = -145.2307692 24.92307692x + x2 = -145.2307692 The x term is 24.92307692x. Take half its coefficient (12.46153846). Square it (155.2899408) and add it to both sides. Add '155.2899408' to each side of the equation. 24.92307692x + 155.2899408 + x2 = -145.2307692 + 155.2899408 Reorder the terms: 155.2899408 + 24.92307692x + x2 = -145.2307692 + 155.2899408 Combine like terms: -145.2307692 + 155.2899408 = 10.0591716 155.2899408 + 24.92307692x + x2 = 10.0591716 Factor a perfect square on the left side: (x + 12.46153846)(x + 12.46153846) = 10.0591716 Calculate the square root of the right side: 3.171619712 Break this problem into two subproblems by setting (x + 12.46153846) equal to 3.171619712 and -3.171619712.Subproblem 1
x + 12.46153846 = 3.171619712 Simplifying x + 12.46153846 = 3.171619712 Reorder the terms: 12.46153846 + x = 3.171619712 Solving 12.46153846 + x = 3.171619712 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.46153846' to each side of the equation. 12.46153846 + -12.46153846 + x = 3.171619712 + -12.46153846 Combine like terms: 12.46153846 + -12.46153846 = 0.00000000 0.00000000 + x = 3.171619712 + -12.46153846 x = 3.171619712 + -12.46153846 Combine like terms: 3.171619712 + -12.46153846 = -9.289918748 x = -9.289918748 Simplifying x = -9.289918748Subproblem 2
x + 12.46153846 = -3.171619712 Simplifying x + 12.46153846 = -3.171619712 Reorder the terms: 12.46153846 + x = -3.171619712 Solving 12.46153846 + x = -3.171619712 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.46153846' to each side of the equation. 12.46153846 + -12.46153846 + x = -3.171619712 + -12.46153846 Combine like terms: 12.46153846 + -12.46153846 = 0.00000000 0.00000000 + x = -3.171619712 + -12.46153846 x = -3.171619712 + -12.46153846 Combine like terms: -3.171619712 + -12.46153846 = -15.633158172 x = -15.633158172 Simplifying x = -15.633158172Solution
The solution to the problem is based on the solutions from the subproblems. x = {-9.289918748, -15.633158172}Solution
x = {-9.289918748, -15.633158172}
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