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Simplifying (3x + 1)(2x + -3) = 1 Reorder the terms: (1 + 3x)(2x + -3) = 1 Reorder the terms: (1 + 3x)(-3 + 2x) = 1 Multiply (1 + 3x) * (-3 + 2x) (1(-3 + 2x) + 3x * (-3 + 2x)) = 1 ((-3 * 1 + 2x * 1) + 3x * (-3 + 2x)) = 1 ((-3 + 2x) + 3x * (-3 + 2x)) = 1 (-3 + 2x + (-3 * 3x + 2x * 3x)) = 1 (-3 + 2x + (-9x + 6x2)) = 1 Combine like terms: 2x + -9x = -7x (-3 + -7x + 6x2) = 1 Solving -3 + -7x + 6x2 = 1 Solving for variable 'x'. Reorder the terms: -3 + -1 + -7x + 6x2 = 1 + -1 Combine like terms: -3 + -1 = -4 -4 + -7x + 6x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -4 + -7x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.6666666667 + -1.166666667x + x2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -1.166666667x + 0.6666666667 + x2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -1.166666667x + x2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -1.166666667x + x2 = 0 + 0.6666666667 -1.166666667x + x2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -1.166666667x + x2 = 0.6666666667 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 = 0.6666666667 + 0.3402777780 Reorder the terms: 0.3402777780 + -1.166666667x + x2 = 0.6666666667 + 0.3402777780 Combine like terms: 0.6666666667 + 0.3402777780 = 1.0069444447 0.3402777780 + -1.166666667x + x2 = 1.0069444447 Factor a perfect square on the left side: (x + -0.5833333335)(x + -0.5833333335) = 1.0069444447 Calculate the square root of the right side: 1.003466215 Break this problem into two subproblems by setting (x + -0.5833333335) equal to 1.003466215 and -1.003466215.Subproblem 1
x + -0.5833333335 = 1.003466215 Simplifying x + -0.5833333335 = 1.003466215 Reorder the terms: -0.5833333335 + x = 1.003466215 Solving -0.5833333335 + x = 1.003466215 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 = 1.003466215 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.003466215 + 0.5833333335 x = 1.003466215 + 0.5833333335 Combine like terms: 1.003466215 + 0.5833333335 = 1.5867995485 x = 1.5867995485 Simplifying x = 1.5867995485Subproblem 2
x + -0.5833333335 = -1.003466215 Simplifying x + -0.5833333335 = -1.003466215 Reorder the terms: -0.5833333335 + x = -1.003466215 Solving -0.5833333335 + x = -1.003466215 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 = -1.003466215 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.003466215 + 0.5833333335 x = -1.003466215 + 0.5833333335 Combine like terms: -1.003466215 + 0.5833333335 = -0.4201328815 x = -0.4201328815 Simplifying x = -0.4201328815Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5867995485, -0.4201328815}
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