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Simplifying (3x + 2)(4x + 3) = 122 Reorder the terms: (2 + 3x)(4x + 3) = 122 Reorder the terms: (2 + 3x)(3 + 4x) = 122 Multiply (2 + 3x) * (3 + 4x) (2(3 + 4x) + 3x * (3 + 4x)) = 122 ((3 * 2 + 4x * 2) + 3x * (3 + 4x)) = 122 ((6 + 8x) + 3x * (3 + 4x)) = 122 (6 + 8x + (3 * 3x + 4x * 3x)) = 122 (6 + 8x + (9x + 12x2)) = 122 Combine like terms: 8x + 9x = 17x (6 + 17x + 12x2) = 122 Solving 6 + 17x + 12x2 = 122 Solving for variable 'x'. Reorder the terms: 6 + -122 + 17x + 12x2 = 122 + -122 Combine like terms: 6 + -122 = -116 -116 + 17x + 12x2 = 122 + -122 Combine like terms: 122 + -122 = 0 -116 + 17x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -9.666666667 + 1.416666667x + x2 = 0 Move the constant term to the right: Add '9.666666667' to each side of the equation. -9.666666667 + 1.416666667x + 9.666666667 + x2 = 0 + 9.666666667 Reorder the terms: -9.666666667 + 9.666666667 + 1.416666667x + x2 = 0 + 9.666666667 Combine like terms: -9.666666667 + 9.666666667 = 0.000000000 0.000000000 + 1.416666667x + x2 = 0 + 9.666666667 1.416666667x + x2 = 0 + 9.666666667 Combine like terms: 0 + 9.666666667 = 9.666666667 1.416666667x + x2 = 9.666666667 The x term is 1.416666667x. Take half its coefficient (0.7083333335). Square it (0.5017361113) and add it to both sides. Add '0.5017361113' to each side of the equation. 1.416666667x + 0.5017361113 + x2 = 9.666666667 + 0.5017361113 Reorder the terms: 0.5017361113 + 1.416666667x + x2 = 9.666666667 + 0.5017361113 Combine like terms: 9.666666667 + 0.5017361113 = 10.1684027783 0.5017361113 + 1.416666667x + x2 = 10.1684027783 Factor a perfect square on the left side: (x + 0.7083333335)(x + 0.7083333335) = 10.1684027783 Calculate the square root of the right side: 3.188793311 Break this problem into two subproblems by setting (x + 0.7083333335) equal to 3.188793311 and -3.188793311.Subproblem 1
x + 0.7083333335 = 3.188793311 Simplifying x + 0.7083333335 = 3.188793311 Reorder the terms: 0.7083333335 + x = 3.188793311 Solving 0.7083333335 + x = 3.188793311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + x = 3.188793311 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + x = 3.188793311 + -0.7083333335 x = 3.188793311 + -0.7083333335 Combine like terms: 3.188793311 + -0.7083333335 = 2.4804599775 x = 2.4804599775 Simplifying x = 2.4804599775Subproblem 2
x + 0.7083333335 = -3.188793311 Simplifying x + 0.7083333335 = -3.188793311 Reorder the terms: 0.7083333335 + x = -3.188793311 Solving 0.7083333335 + x = -3.188793311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + x = -3.188793311 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + x = -3.188793311 + -0.7083333335 x = -3.188793311 + -0.7083333335 Combine like terms: -3.188793311 + -0.7083333335 = -3.8971266445 x = -3.8971266445 Simplifying x = -3.8971266445Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4804599775, -3.8971266445}
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