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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 = 15x 64(-100 + 15x + x2) + 4(x + 12)(x + -4) = 16(x + -4)(x + 16) (-100 * 64 + 15x * 64 + x2 * 64) + 4(x + 12)(x + -4) = 16(x + -4)(x + 16) (-6400 + 960x + 64x2) + 4(x + 12)(x + -4) = 16(x + -4)(x + 16) Reorder the terms: -6400 + 960x + 64x2 + 4(12 + x)(x + -4) = 16(x + -4)(x + 16) Reorder the terms: -6400 + 960x + 64x2 + 4(12 + x)(-4 + x) = 16(x + -4)(x + 16) Multiply (12 + x) * (-4 + x) -6400 + 960x + 64x2 + 4(12(-4 + x) + x(-4 + x)) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + 4((-4 * 12 + x * 12) + x(-4 + x)) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + 4((-48 + 12x) + x(-4 + x)) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + 4(-48 + 12x + (-4 * x + x * x)) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + 4(-48 + 12x + (-4x + x2)) = 16(x + -4)(x + 16) Combine like terms: 12x + -4x = 8x -6400 + 960x + 64x2 + 4(-48 + 8x + x2) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + (-48 * 4 + 8x * 4 + x2 * 4) = 16(x + -4)(x + 16) -6400 + 960x + 64x2 + (-192 + 32x + 4x2) = 16(x + -4)(x + 16) Reorder the terms: -6400 + -192 + 960x + 32x + 64x2 + 4x2 = 16(x + -4)(x + 16) Combine like terms: -6400 + -192 = -6592 -6592 + 960x + 32x + 64x2 + 4x2 = 16(x + -4)(x + 16) Combine like terms: 960x + 32x = 992x -6592 + 992x + 64x2 + 4x2 = 16(x + -4)(x + 16) Combine like terms: 64x2 + 4x2 = 68x2 -6592 + 992x + 68x2 = 16(x + -4)(x + 16) Reorder the terms: -6592 + 992x + 68x2 = 16(-4 + x)(x + 16) Reorder the terms: -6592 + 992x + 68x2 = 16(-4 + x)(16 + x) Multiply (-4 + x) * (16 + x) -6592 + 992x + 68x2 = 16(-4(16 + x) + x(16 + x)) -6592 + 992x + 68x2 = 16((16 * -4 + x * -4) + x(16 + x)) -6592 + 992x + 68x2 = 16((-64 + -4x) + x(16 + x)) -6592 + 992x + 68x2 = 16(-64 + -4x + (16 * x + x * x)) -6592 + 992x + 68x2 = 16(-64 + -4x + (16x + x2)) Combine like terms: -4x + 16x = 12x -6592 + 992x + 68x2 = 16(-64 + 12x + x2) -6592 + 992x + 68x2 = (-64 * 16 + 12x * 16 + x2 * 16) -6592 + 992x + 68x2 = (-1024 + 192x + 16x2) Solving -6592 + 992x + 68x2 = -1024 + 192x + 16x2 Solving for variable 'x'. Reorder the terms: -6592 + 1024 + 992x + -192x + 68x2 + -16x2 = -1024 + 192x + 16x2 + 1024 + -192x + -16x2 Combine like terms: -6592 + 1024 = -5568 -5568 + 992x + -192x + 68x2 + -16x2 = -1024 + 192x + 16x2 + 1024 + -192x + -16x2 Combine like terms: 992x + -192x = 800x -5568 + 800x + 68x2 + -16x2 = -1024 + 192x + 16x2 + 1024 + -192x + -16x2 Combine like terms: 68x2 + -16x2 = 52x2 -5568 + 800x + 52x2 = -1024 + 192x + 16x2 + 1024 + -192x + -16x2 Reorder the terms: -5568 + 800x + 52x2 = -1024 + 1024 + 192x + -192x + 16x2 + -16x2 Combine like terms: -1024 + 1024 = 0 -5568 + 800x + 52x2 = 0 + 192x + -192x + 16x2 + -16x2 -5568 + 800x + 52x2 = 192x + -192x + 16x2 + -16x2 Combine like terms: 192x + -192x = 0 -5568 + 800x + 52x2 = 0 + 16x2 + -16x2 -5568 + 800x + 52x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 -5568 + 800x + 52x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1392 + 200x + 13x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1392 + 200x + 13x2)' equal to zero and attempt to solve: Simplifying -1392 + 200x + 13x2 = 0 Solving -1392 + 200x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -107.0769231 + 15.38461538x + x2 = 0 Move the constant term to the right: Add '107.0769231' to each side of the equation. -107.0769231 + 15.38461538x + 107.0769231 + x2 = 0 + 107.0769231 Reorder the terms: -107.0769231 + 107.0769231 + 15.38461538x + x2 = 0 + 107.0769231 Combine like terms: -107.0769231 + 107.0769231 = 0.0000000 0.0000000 + 15.38461538x + x2 = 0 + 107.0769231 15.38461538x + x2 = 0 + 107.0769231 Combine like terms: 0 + 107.0769231 = 107.0769231 15.38461538x + x2 = 107.0769231 The x term is 15.38461538x. Take half its coefficient (7.69230769). Square it (59.17159760) and add it to both sides. Add '59.17159760' to each side of the equation. 15.38461538x + 59.17159760 + x2 = 107.0769231 + 59.17159760 Reorder the terms: 59.17159760 + 15.38461538x + x2 = 107.0769231 + 59.17159760 Combine like terms: 107.0769231 + 59.17159760 = 166.2485207 59.17159760 + 15.38461538x + x2 = 166.2485207 Factor a perfect square on the left side: (x + 7.69230769)(x + 7.69230769) = 166.2485207 Calculate the square root of the right side: 12.893739593 Break this problem into two subproblems by setting (x + 7.69230769) equal to 12.893739593 and -12.893739593.Subproblem 1
x + 7.69230769 = 12.893739593 Simplifying x + 7.69230769 = 12.893739593 Reorder the terms: 7.69230769 + x = 12.893739593 Solving 7.69230769 + x = 12.893739593 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.69230769' to each side of the equation. 7.69230769 + -7.69230769 + x = 12.893739593 + -7.69230769 Combine like terms: 7.69230769 + -7.69230769 = 0.00000000 0.00000000 + x = 12.893739593 + -7.69230769 x = 12.893739593 + -7.69230769 Combine like terms: 12.893739593 + -7.69230769 = 5.201431903 x = 5.201431903 Simplifying x = 5.201431903Subproblem 2
x + 7.69230769 = -12.893739593 Simplifying x + 7.69230769 = -12.893739593 Reorder the terms: 7.69230769 + x = -12.893739593 Solving 7.69230769 + x = -12.893739593 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.69230769' to each side of the equation. 7.69230769 + -7.69230769 + x = -12.893739593 + -7.69230769 Combine like terms: 7.69230769 + -7.69230769 = 0.00000000 0.00000000 + x = -12.893739593 + -7.69230769 x = -12.893739593 + -7.69230769 Combine like terms: -12.893739593 + -7.69230769 = -20.586047283 x = -20.586047283 Simplifying x = -20.586047283Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.201431903, -20.586047283}Solution
x = {5.201431903, -20.586047283}
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