(3a-1)*(2a+3)-(4a-2)*(3a+5)=

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Solution for (3a-1)*(2a+3)-(4a-2)*(3a+5)= equation:


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 = 0.6442433215

Subproblem 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 = -1.8109099885

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.6442433215, -1.8109099885}

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