Determine Variables A, B, and C for the Equation: (A/X) + (BX/2) = ((2X+3)²)/X - C

To solve this problem, we will simplify both sides of the given equation:

Given equation:

$\frac{A}{X} + \frac{BX}{2} = \frac{(2X+3)^2}{X} - C$.

First, expand the quadratic expression:

$(2X+3)^2 = (2X+3)(2X+3) = 4X^2 + 6X + 6X + 9 = 4X^2 + 12X + 9$.

Substitute this back into the equation:

$\frac{A}{X} + \frac{BX}{2} = \frac{4X^2 + 12X + 9}{X} - C$.

Simplify the right-hand side:

$\frac{4X^2 + 12X + 9}{X} = 4X + \frac{12X}{X} + \frac{9}{X} = 4X + 12 + \frac{9}{X}$.

The equation now becomes:

$\frac{A}{X} + \frac{BX}{2} = 4X + 12 + \frac{9}{X} - C$.

For the equation to hold true for all values of $X$, equate corresponding terms:

• $\frac{A}{X} = \frac{9}{X} \Rightarrow A = 9$.
• $\frac{BX}{2} = 4X \Rightarrow B = 8$.
• For constant terms: $12 - C = 0 \Rightarrow C = 12$.

Therefore, the values are $A = 9$, $B = 8$, and $C = 12$.

The correct answer is: $A = 9, B = 8, C = 12$.