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

Video Solution

Solution Steps

00:00 Find A,B,C

00:03 Multiply by the common denominator to eliminate fractions

00:44 Reduce what's possible

00:56 Use shortened multiplication formulas to open the parentheses

01:19 Move C to the left side

01:24 Calculate the squares and products

01:40 Open parentheses properly, multiply by each factor

01:50 Equate each coefficient on the right side to the coefficient on the left side

01:56 This is the solution for B

02:07 Factor 24 into factors 2 and 12

02:16 This is the solution for C

02:23 Factor 18 into factors 9 and 2

02:27 This is the solution for A

02:30 And this is the solution to the question

Step-by-Step Solution

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:

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

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