Simplify $\frac{324 - 4x^2}{2x + 18}$

To simplify the expression $\frac{324 - 4x^2}{2x + 18}$, we will use algebraic factorization techniques.

1. Analyze the expression

The numerator is $324 - 4x^2$. Notice that both terms are perfect squares:

• $324 = 18^2$
• $4x^2 = (2x)^2$

This fits the Difference of Two Squares formula: $a^2 - b^2 = (a - b)(a + b)$.

2. Factorize the numerator

Applying the formula where $a = 18$ and $b = 2x$:

Alternatively, we can factor out the common factor of $4$ first:

3. Factorize the denominator

The denominator is $2x + 18$. We can factor out the common factor of $2$:

Since addition is commutative, $2(x + 9)$ is the same as $2(9 + x)$.

4. Simplify the fraction

Now, substitute the factored forms back into the original expression:

Cancel out the common term $(9 + x)$ from the numerator and the denominator:

Divide $4$ by $2$:

5. Compare with the options

We have $2(9 - x)$. Let's look at the options provided:

• A. $2(x - 9)$
• B. $2(9 + x)$
• C. $81 - x^2$
• D. $-2(x - 9)$

Note that $2(9 - x)$ is mathematically equivalent to $-2(x - 9)$ because: $-2 \times x = -2x$ $-2 \times -9 = +18$ So, $-2(x - 9) = 18 - 2x$, which is the same as $2(9 - x)$.

Exam Tip: When you see a negative sign inside a bracket, remember that $-(a - b)$ is the same as $(b - a)$. Always check if your result can be rearranged to match the options by factoring out a $-1$.