Solve Complex Fraction Equation: Finding A, B, C in (A/x + Bx/2) = (2x-3)²/x - C

Video Solution

Solution Steps

00:15 Let's find A, B, and C that make the equation true.

00:20 First, multiply by the common denominator to get rid of fractions.

00:49 Now, rearrange the equation terms neatly.

00:55 Use the multiplication formulas to expand any brackets.

01:13 Next, calculate the squares and products carefully.

01:27 Expand each bracket by multiplying with the factors.

01:37 Compare like terms to figure out A, B, and C.

01:43 Here's the solution for B.

01:49 Use the same method to find values for C and A.

01:58 Now, this is the solution for C.

02:06 And finally, here's the solution for A.

02:10 Great job! That's how we solve the problem!

Step-by-Step Solution

To solve this problem, we'll proceed with the following steps:

Step 1: Expand the expression $(2x-3)^2$ on the right-hand side.
Step 2: Simplify the resulting expression by dividing by $x$ .
Step 3: Compare the terms on both sides of the equation to deduce the coefficients $A$ , $B$ , and $C$ .

Now, let us work through these steps:

Step 1: Expand the expression $(2x-3)^2$ .

$(2x - 3)^2 = 4x^2 - 12x + 9$

Step 2: Substitute back into the equation and simplify by dividing by $x$ :

$\frac{A}{x} + \frac{Bx}{2} = \frac{4x^2 - 12x + 9}{x} - C$

This simplifies to:

$4x - 12 + \frac{9}{x} - C$

Step 3: To find $A$ , $B$ , and $C$ , compare coefficients of similar terms:

Constant term (no $x$ ): Equate $-12$ with $-C$ to get $C = 12$ (note: should consider sign reversal).
Coefficient of $x$ : Compare $\frac{B}{2}$ to 4. Giving $B = 8$ .
Coefficient of $\frac{1}{x}$ : Compare $A$ to 9, giving $A = 9$ .

Therefore, the solution to the problem is that the values are $\textbf{A = 9, B = 8, C = -12}$ .