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Simplifying ((1x + 8)(1x + -2)) + ((1x + 3)(1x + 5)) = 123 Reorder the terms: ((8 + 1x)(1x + -2)) + ((1x + 3)(1x + 5)) = 123 Reorder the terms: ((8 + 1x)(-2 + 1x)) + ((1x + 3)(1x + 5)) = 123 Multiply (8 + 1x) * (-2 + 1x) ((8(-2 + 1x) + 1x * (-2 + 1x))) + ((1x + 3)(1x + 5)) = 123 (((-2 * 8 + 1x * 8) + 1x * (-2 + 1x))) + ((1x + 3)(1x + 5)) = 123 (((-16 + 8x) + 1x * (-2 + 1x))) + ((1x + 3)(1x + 5)) = 123 ((-16 + 8x + (-2 * 1x + 1x * 1x))) + ((1x + 3)(1x + 5)) = 123 ((-16 + 8x + (-2x + 1x2))) + ((1x + 3)(1x + 5)) = 123 Combine like terms: 8x + -2x = 6x ((-16 + 6x + 1x2)) + ((1x + 3)(1x + 5)) = 123 (-16 + 6x + 1x2) + ((1x + 3)(1x + 5)) = 123 Remove parenthesis around (-16 + 6x + 1x2) -16 + 6x + 1x2 + ((1x + 3)(1x + 5)) = 123 Reorder the terms: -16 + 6x + 1x2 + ((3 + 1x)(1x + 5)) = 123 Reorder the terms: -16 + 6x + 1x2 + ((3 + 1x)(5 + 1x)) = 123 Multiply (3 + 1x) * (5 + 1x) -16 + 6x + 1x2 + ((3(5 + 1x) + 1x * (5 + 1x))) = 123 -16 + 6x + 1x2 + (((5 * 3 + 1x * 3) + 1x * (5 + 1x))) = 123 -16 + 6x + 1x2 + (((15 + 3x) + 1x * (5 + 1x))) = 123 -16 + 6x + 1x2 + ((15 + 3x + (5 * 1x + 1x * 1x))) = 123 -16 + 6x + 1x2 + ((15 + 3x + (5x + 1x2))) = 123 Combine like terms: 3x + 5x = 8x -16 + 6x + 1x2 + ((15 + 8x + 1x2)) = 123 -16 + 6x + 1x2 + (15 + 8x + 1x2) = 123 Remove parenthesis around (15 + 8x + 1x2) -16 + 6x + 1x2 + 15 + 8x + 1x2 = 123 Reorder the terms: -16 + 15 + 6x + 8x + 1x2 + 1x2 = 123 Combine like terms: -16 + 15 = -1 -1 + 6x + 8x + 1x2 + 1x2 = 123 Combine like terms: 6x + 8x = 14x -1 + 14x + 1x2 + 1x2 = 123 Combine like terms: 1x2 + 1x2 = 2x2 -1 + 14x + 2x2 = 123 Solving -1 + 14x + 2x2 = 123 Solving for variable 'x'. Reorder the terms: -1 + -123 + 14x + 2x2 = 123 + -123 Combine like terms: -1 + -123 = -124 -124 + 14x + 2x2 = 123 + -123 Combine like terms: 123 + -123 = 0 -124 + 14x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-62 + 7x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-62 + 7x + x2)' equal to zero and attempt to solve: Simplifying -62 + 7x + x2 = 0 Solving -62 + 7x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '62' to each side of the equation. -62 + 7x + 62 + x2 = 0 + 62 Reorder the terms: -62 + 62 + 7x + x2 = 0 + 62 Combine like terms: -62 + 62 = 0 0 + 7x + x2 = 0 + 62 7x + x2 = 0 + 62 Combine like terms: 0 + 62 = 62 7x + x2 = 62 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 62 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 62 + 12.25 Combine like terms: 62 + 12.25 = 74.25 12.25 + 7x + x2 = 74.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 74.25 Calculate the square root of the right side: 8.61684397 Break this problem into two subproblems by setting (x + 3.5) equal to 8.61684397 and -8.61684397.Subproblem 1
x + 3.5 = 8.61684397 Simplifying x + 3.5 = 8.61684397 Reorder the terms: 3.5 + x = 8.61684397 Solving 3.5 + x = 8.61684397 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 8.61684397 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 8.61684397 + -3.5 x = 8.61684397 + -3.5 Combine like terms: 8.61684397 + -3.5 = 5.11684397 x = 5.11684397 Simplifying x = 5.11684397Subproblem 2
x + 3.5 = -8.61684397 Simplifying x + 3.5 = -8.61684397 Reorder the terms: 3.5 + x = -8.61684397 Solving 3.5 + x = -8.61684397 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -8.61684397 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -8.61684397 + -3.5 x = -8.61684397 + -3.5 Combine like terms: -8.61684397 + -3.5 = -12.11684397 x = -12.11684397 Simplifying x = -12.11684397Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.11684397, -12.11684397}Solution
x = {5.11684397, -12.11684397}
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