Final answer:
The sum of the roots of the equation (2x+3)(x-4) + (2x+3)(x-6) = 0 is 7/2 or 3.5, after factoring out the common term and finding the roots of the resulting quadratic equation. None of the provided answer choices match this result.
Step-by-step explanation:
To find the sum of all the roots of the equation (2x+3)(x-4) + (2x+3)(x-6) = 0, first we can factor out the common term (2x+3):
(2x + 3) [(x - 4) + (x - 6)] = 0
(2x + 3) (x - 4 + x - 6) = 0
(2x + 3) (2x - 10) = 0
Now we have a product of two factors equal to zero, which tells us that one or both of the factors must be zero. This gives us two separate equations to solve for x:
2x + 3 = 0 or 2x - 10 = 0
To find the roots, we solve each equation separately:
2x + 3 = 0 → x = -3/2
2x - 10 = 0 → x = 10/2
So the roots of the equation are x = -3/2 and x = 5. The sum of these roots is (-3/2) + 5.
To calculate the sum, convert 5 to a fraction with a denominator of 2: 5 = 10/2.
The sum of the roots is: (-3/2) + (10/2) = (10 - 3)/2 = 7/2
The answer is therefore 7/2 or 3.5, which is not listed among the options provided (a) 1, (b) 2, (c) 3, (d) 4, meaning there may be an error in the question or the answer choices.