1Cademy - Simplifying $$\frac{\frac{4}{m^2-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}$$ Using the LCD

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Simplifying a Complex Rational Expression by Using the LCD

Example

Simplifying $\frac{\frac{4}{m^2-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}$ Using the LCD

Simplify the complex rational expression using the LCD method:

$\frac{\frac{4}{m^2-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}$

Step 1 — Find the LCD of all inner fractions. The inner denominators are $m^2-7m+12$ , $(m-3)$ , and $(m-4)$ . Factor the trinomial: $m^2-7m+12 = (m-3)(m-4)$ . Since the factored form already contains both of the other denominators, the LCD is $(m-3)(m-4)$ .

Step 2 — Multiply the numerator and denominator of the complex fraction by the LCD $(m-3)(m-4)$ . Distribute to each term:

$\frac{(m-3)(m-4) \cdot \frac{4}{(m-3)(m-4)}}{(m-3)(m-4) \cdot \frac{3}{m-3} - (m-3)(m-4) \cdot \frac{2}{m-4}}$

Step 3 — Simplify by canceling matching factors. In the numerator, $(m-3)(m-4)$ cancels completely, leaving $4$ . In the first denominator term, $(m-3)$ cancels, giving $3(m-4)$ . In the second, $(m-4)$ cancels, giving $2(m-3)$ :

$\frac{4}{3(m-4) - 2(m-3)}$

Step 4 — Distribute in the denominator: $3(m-4) = 3m - 12$ and $2(m-3) = 2m - 6$ :

$\frac{4}{3m - 12 - 2m + 6}$

Step 5 — Combine like terms: $3m - 2m = m$ and $-12 + 6 = -6$ :

$\frac{4}{m - 6}$

The expression is fully simplified. This example illustrates the LCD method when the numerator of the complex rational expression contains a single fraction whose denominator is a factorable trinomial. Factoring $m^2-7m+12$ into $(m-3)(m-4)$ reveals that the LCD equals this trinomial's factored form — the same product that already appears as the two individual denominators in the denominator of the complex fraction. After multiplying by the LCD, all inner fractions clear at once, and the remaining algebra involves only distributing and combining like terms.

Updated 2026-04-30

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