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Simplifying (3a + -1)(2a + 3) + -1(4a + -2)(3a + 5) = 0 Reorder the terms: (-1 + 3a)(2a + 3) + -1(4a + -2)(3a + 5) = 0 Reorder the terms: (-1 + 3a)(3 + 2a) + -1(4a + -2)(3a + 5) = 0 Multiply (-1 + 3a) * (3 + 2a) (-1(3 + 2a) + 3a * (3 + 2a)) + -1(4a + -2)(3a + 5) = 0 ((3 * -1 + 2a * -1) + 3a * (3 + 2a)) + -1(4a + -2)(3a + 5) = 0 ((-3 + -2a) + 3a * (3 + 2a)) + -1(4a + -2)(3a + 5) = 0 (-3 + -2a + (3 * 3a + 2a * 3a)) + -1(4a + -2)(3a + 5) = 0 (-3 + -2a + (9a + 6a2)) + -1(4a + -2)(3a + 5) = 0 Combine like terms: -2a + 9a = 7a (-3 + 7a + 6a2) + -1(4a + -2)(3a + 5) = 0 Reorder the terms: -3 + 7a + 6a2 + -1(-2 + 4a)(3a + 5) = 0 Reorder the terms: -3 + 7a + 6a2 + -1(-2 + 4a)(5 + 3a) = 0 Multiply (-2 + 4a) * (5 + 3a) -3 + 7a + 6a2 + -1(-2(5 + 3a) + 4a * (5 + 3a)) = 0 -3 + 7a + 6a2 + -1((5 * -2 + 3a * -2) + 4a * (5 + 3a)) = 0 -3 + 7a + 6a2 + -1((-10 + -6a) + 4a * (5 + 3a)) = 0 -3 + 7a + 6a2 + -1(-10 + -6a + (5 * 4a + 3a * 4a)) = 0 -3 + 7a + 6a2 + -1(-10 + -6a + (20a + 12a2)) = 0 Combine like terms: -6a + 20a = 14a -3 + 7a + 6a2 + -1(-10 + 14a + 12a2) = 0 -3 + 7a + 6a2 + (-10 * -1 + 14a * -1 + 12a2 * -1) = 0 -3 + 7a + 6a2 + (10 + -14a + -12a2) = 0 Reorder the terms: -3 + 10 + 7a + -14a + 6a2 + -12a2 = 0 Combine like terms: -3 + 10 = 7 7 + 7a + -14a + 6a2 + -12a2 = 0 Combine like terms: 7a + -14a = -7a 7 + -7a + 6a2 + -12a2 = 0 Combine like terms: 6a2 + -12a2 = -6a2 7 + -7a + -6a2 = 0 Solving 7 + -7a + -6a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.166666667 + 1.166666667a + a2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667a + 1.166666667 + a2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 1.166666667a + a2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 1.166666667a + a2 = 0 + 1.166666667 1.166666667a + a2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 1.166666667a + a2 = 1.166666667 The a term is 1.166666667a. 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. 1.166666667a + 0.3402777780 + a2 = 1.166666667 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667a + a2 = 1.166666667 + 0.3402777780 Combine like terms: 1.166666667 + 0.3402777780 = 1.506944445 0.3402777780 + 1.166666667a + a2 = 1.506944445 Factor a perfect square on the left side: (a + 0.5833333335)(a + 0.5833333335) = 1.506944445 Calculate the square root of the right side: 1.227576655 Break this problem into two subproblems by setting (a + 0.5833333335) equal to 1.227576655 and -1.227576655.Subproblem 1
a + 0.5833333335 = 1.227576655 Simplifying a + 0.5833333335 = 1.227576655 Reorder the terms: 0.5833333335 + a = 1.227576655 Solving 0.5833333335 + a = 1.227576655 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + a = 1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + a = 1.227576655 + -0.5833333335 a = 1.227576655 + -0.5833333335 Combine like terms: 1.227576655 + -0.5833333335 = 0.6442433215 a = 0.6442433215 Simplifying a = 0.6442433215Subproblem 2
a + 0.5833333335 = -1.227576655 Simplifying a + 0.5833333335 = -1.227576655 Reorder the terms: 0.5833333335 + a = -1.227576655 Solving 0.5833333335 + a = -1.227576655 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + a = -1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + a = -1.227576655 + -0.5833333335 a = -1.227576655 + -0.5833333335 Combine like terms: -1.227576655 + -0.5833333335 = -1.8109099885 a = -1.8109099885 Simplifying a = -1.8109099885Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.6442433215, -1.8109099885}
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