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Simplifying (3a + -1) * x(2a + 3) + -1(4a + -2) * x(3a + 5) = 0 Reorder the terms: (-1 + 3a) * x(2a + 3) + -1(4a + -2) * x(3a + 5) = 0 Reorder the terms: (-1 + 3a) * x(3 + 2a) + -1(4a + -2) * x(3a + 5) = 0 Reorder the terms for easier multiplication: x(-1 + 3a)(3 + 2a) + -1(4a + -2) * x(3a + 5) = 0 Multiply (-1 + 3a) * (3 + 2a) x(-1(3 + 2a) + 3a * (3 + 2a)) + -1(4a + -2) * x(3a + 5) = 0 x((3 * -1 + 2a * -1) + 3a * (3 + 2a)) + -1(4a + -2) * x(3a + 5) = 0 x((-3 + -2a) + 3a * (3 + 2a)) + -1(4a + -2) * x(3a + 5) = 0 x(-3 + -2a + (3 * 3a + 2a * 3a)) + -1(4a + -2) * x(3a + 5) = 0 x(-3 + -2a + (9a + 6a2)) + -1(4a + -2) * x(3a + 5) = 0 Combine like terms: -2a + 9a = 7a x(-3 + 7a + 6a2) + -1(4a + -2) * x(3a + 5) = 0 (-3 * x + 7a * x + 6a2 * x) + -1(4a + -2) * x(3a + 5) = 0 Reorder the terms: (7ax + 6a2x + -3x) + -1(4a + -2) * x(3a + 5) = 0 (7ax + 6a2x + -3x) + -1(4a + -2) * x(3a + 5) = 0 Reorder the terms: 7ax + 6a2x + -3x + -1(-2 + 4a) * x(3a + 5) = 0 Reorder the terms: 7ax + 6a2x + -3x + -1(-2 + 4a) * x(5 + 3a) = 0 Reorder the terms for easier multiplication: 7ax + 6a2x + -3x + -1x(-2 + 4a)(5 + 3a) = 0 Multiply (-2 + 4a) * (5 + 3a) 7ax + 6a2x + -3x + -1x(-2(5 + 3a) + 4a * (5 + 3a)) = 0 7ax + 6a2x + -3x + -1x((5 * -2 + 3a * -2) + 4a * (5 + 3a)) = 0 7ax + 6a2x + -3x + -1x((-10 + -6a) + 4a * (5 + 3a)) = 0 7ax + 6a2x + -3x + -1x(-10 + -6a + (5 * 4a + 3a * 4a)) = 0 7ax + 6a2x + -3x + -1x(-10 + -6a + (20a + 12a2)) = 0 Combine like terms: -6a + 20a = 14a 7ax + 6a2x + -3x + -1x(-10 + 14a + 12a2) = 0 7ax + 6a2x + -3x + (-10 * -1x + 14a * -1x + 12a2 * -1x) = 0 Reorder the terms: 7ax + 6a2x + -3x + (-14ax + -12a2x + 10x) = 0 7ax + 6a2x + -3x + (-14ax + -12a2x + 10x) = 0 Reorder the terms: 7ax + -14ax + 6a2x + -12a2x + -3x + 10x = 0 Combine like terms: 7ax + -14ax = -7ax -7ax + 6a2x + -12a2x + -3x + 10x = 0 Combine like terms: 6a2x + -12a2x = -6a2x -7ax + -6a2x + -3x + 10x = 0 Combine like terms: -3x + 10x = 7x -7ax + -6a2x + 7x = 0 Solving -7ax + -6a2x + 7x = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'x'. x(-7a + -6a2 + 7) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing a to the left, all other terms to the right. Add '-1x' to each side of the equation. x + -1x = 0 + -1x Remove the zero: 0 = -1x Simplifying 0 = -1x The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(-7a + -6a2 + 7)' equal to zero and attempt to solve: Simplifying -7a + -6a2 + 7 = 0 Reorder the terms: 7 + -7a + -6a2 = 0 Solving 7 + -7a + -6a2 = 0 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}Solution
a = {0.6442433215, -1.8109099885}
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