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Simplifying 24 + (12 + -5x)(3 + x) = 22 + (12 + 2x)(2 + -1x) Multiply (12 + -5x) * (3 + x) 24 + (12(3 + x) + -5x * (3 + x)) = 22 + (12 + 2x)(2 + -1x) 24 + ((3 * 12 + x * 12) + -5x * (3 + x)) = 22 + (12 + 2x)(2 + -1x) 24 + ((36 + 12x) + -5x * (3 + x)) = 22 + (12 + 2x)(2 + -1x) 24 + (36 + 12x + (3 * -5x + x * -5x)) = 22 + (12 + 2x)(2 + -1x) 24 + (36 + 12x + (-15x + -5x2)) = 22 + (12 + 2x)(2 + -1x) Combine like terms: 12x + -15x = -3x 24 + (36 + -3x + -5x2) = 22 + (12 + 2x)(2 + -1x) Combine like terms: 24 + 36 = 60 60 + -3x + -5x2 = 22 + (12 + 2x)(2 + -1x) Multiply (12 + 2x) * (2 + -1x) 60 + -3x + -5x2 = 22 + (12(2 + -1x) + 2x * (2 + -1x)) 60 + -3x + -5x2 = 22 + ((2 * 12 + -1x * 12) + 2x * (2 + -1x)) 60 + -3x + -5x2 = 22 + ((24 + -12x) + 2x * (2 + -1x)) 60 + -3x + -5x2 = 22 + (24 + -12x + (2 * 2x + -1x * 2x)) 60 + -3x + -5x2 = 22 + (24 + -12x + (4x + -2x2)) Combine like terms: -12x + 4x = -8x 60 + -3x + -5x2 = 22 + (24 + -8x + -2x2) Combine like terms: 22 + 24 = 46 60 + -3x + -5x2 = 46 + -8x + -2x2 Solving 60 + -3x + -5x2 = 46 + -8x + -2x2 Solving for variable 'x'. Reorder the terms: 60 + -46 + -3x + 8x + -5x2 + 2x2 = 46 + -8x + -2x2 + -46 + 8x + 2x2 Combine like terms: 60 + -46 = 14 14 + -3x + 8x + -5x2 + 2x2 = 46 + -8x + -2x2 + -46 + 8x + 2x2 Combine like terms: -3x + 8x = 5x 14 + 5x + -5x2 + 2x2 = 46 + -8x + -2x2 + -46 + 8x + 2x2 Combine like terms: -5x2 + 2x2 = -3x2 14 + 5x + -3x2 = 46 + -8x + -2x2 + -46 + 8x + 2x2 Reorder the terms: 14 + 5x + -3x2 = 46 + -46 + -8x + 8x + -2x2 + 2x2 Combine like terms: 46 + -46 = 0 14 + 5x + -3x2 = 0 + -8x + 8x + -2x2 + 2x2 14 + 5x + -3x2 = -8x + 8x + -2x2 + 2x2 Combine like terms: -8x + 8x = 0 14 + 5x + -3x2 = 0 + -2x2 + 2x2 14 + 5x + -3x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 14 + 5x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -4.666666667 + -1.666666667x + x2 = 0 Move the constant term to the right: Add '4.666666667' to each side of the equation. -4.666666667 + -1.666666667x + 4.666666667 + x2 = 0 + 4.666666667 Reorder the terms: -4.666666667 + 4.666666667 + -1.666666667x + x2 = 0 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + -1.666666667x + x2 = 0 + 4.666666667 -1.666666667x + x2 = 0 + 4.666666667 Combine like terms: 0 + 4.666666667 = 4.666666667 -1.666666667x + x2 = 4.666666667 The x term is -1.666666667x. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667x + 0.6944444447 + x2 = 4.666666667 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667x + x2 = 4.666666667 + 0.6944444447 Combine like terms: 4.666666667 + 0.6944444447 = 5.3611111117 0.6944444447 + -1.666666667x + x2 = 5.3611111117 Factor a perfect square on the left side: (x + -0.8333333335)(x + -0.8333333335) = 5.3611111117 Calculate the square root of the right side: 2.315407332 Break this problem into two subproblems by setting (x + -0.8333333335) equal to 2.315407332 and -2.315407332.Subproblem 1
x + -0.8333333335 = 2.315407332 Simplifying x + -0.8333333335 = 2.315407332 Reorder the terms: -0.8333333335 + x = 2.315407332 Solving -0.8333333335 + x = 2.315407332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.315407332 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.315407332 + 0.8333333335 x = 2.315407332 + 0.8333333335 Combine like terms: 2.315407332 + 0.8333333335 = 3.1487406655 x = 3.1487406655 Simplifying x = 3.1487406655Subproblem 2
x + -0.8333333335 = -2.315407332 Simplifying x + -0.8333333335 = -2.315407332 Reorder the terms: -0.8333333335 + x = -2.315407332 Solving -0.8333333335 + x = -2.315407332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.315407332 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.315407332 + 0.8333333335 x = -2.315407332 + 0.8333333335 Combine like terms: -2.315407332 + 0.8333333335 = -1.4820739985 x = -1.4820739985 Simplifying x = -1.4820739985Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.1487406655, -1.4820739985}
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