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Simplifying (2x + 1)(3x + 2) = 24 Reorder the terms: (1 + 2x)(3x + 2) = 24 Reorder the terms: (1 + 2x)(2 + 3x) = 24 Multiply (1 + 2x) * (2 + 3x) (1(2 + 3x) + 2x * (2 + 3x)) = 24 ((2 * 1 + 3x * 1) + 2x * (2 + 3x)) = 24 ((2 + 3x) + 2x * (2 + 3x)) = 24 (2 + 3x + (2 * 2x + 3x * 2x)) = 24 (2 + 3x + (4x + 6x2)) = 24 Combine like terms: 3x + 4x = 7x (2 + 7x + 6x2) = 24 Solving 2 + 7x + 6x2 = 24 Solving for variable 'x'. Reorder the terms: 2 + -24 + 7x + 6x2 = 24 + -24 Combine like terms: 2 + -24 = -22 -22 + 7x + 6x2 = 24 + -24 Combine like terms: 24 + -24 = 0 -22 + 7x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.666666667 + 1.166666667x + x2 = 0 Move the constant term to the right: Add '3.666666667' to each side of the equation. -3.666666667 + 1.166666667x + 3.666666667 + x2 = 0 + 3.666666667 Reorder the terms: -3.666666667 + 3.666666667 + 1.166666667x + x2 = 0 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + 1.166666667x + x2 = 0 + 3.666666667 1.166666667x + x2 = 0 + 3.666666667 Combine like terms: 0 + 3.666666667 = 3.666666667 1.166666667x + x2 = 3.666666667 The x term is 1.166666667x. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667x + 0.3402777780 + x2 = 3.666666667 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667x + x2 = 3.666666667 + 0.3402777780 Combine like terms: 3.666666667 + 0.3402777780 = 4.006944445 0.3402777780 + 1.166666667x + x2 = 4.006944445 Factor a perfect square on the left side: (x + 0.5833333335)(x + 0.5833333335) = 4.006944445 Calculate the square root of the right side: 2.001735358 Break this problem into two subproblems by setting (x + 0.5833333335) equal to 2.001735358 and -2.001735358.Subproblem 1
x + 0.5833333335 = 2.001735358 Simplifying x + 0.5833333335 = 2.001735358 Reorder the terms: 0.5833333335 + x = 2.001735358 Solving 0.5833333335 + x = 2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = 2.001735358 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = 2.001735358 + -0.5833333335 x = 2.001735358 + -0.5833333335 Combine like terms: 2.001735358 + -0.5833333335 = 1.4184020245 x = 1.4184020245 Simplifying x = 1.4184020245Subproblem 2
x + 0.5833333335 = -2.001735358 Simplifying x + 0.5833333335 = -2.001735358 Reorder the terms: 0.5833333335 + x = -2.001735358 Solving 0.5833333335 + x = -2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = -2.001735358 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = -2.001735358 + -0.5833333335 x = -2.001735358 + -0.5833333335 Combine like terms: -2.001735358 + -0.5833333335 = -2.5850686915 x = -2.5850686915 Simplifying x = -2.5850686915Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4184020245, -2.5850686915}
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