Question

subtract: \\(\frac{3q^2 + q}{9q^2 - 4} - \frac{2}{9q^2 - 4}\\).

Explanation:

Combine the numerators over the common denominator

The given expression is:

$$ \frac{3q^2 + q}{9q^2 - 4} - \frac{2}{9q^2 - 4} $$

Since the denominators are identical, we can perform the subtraction by subtracting the numerators directly, using our understanding of Adding Rational Expressions:

$$ \frac{3q^2 + q - 2}{9q^2 - 4} $$

Factor the numerator

Next, we factor the quadratic trinomial in the numerator, \(3q^2 + q - 2\), using Factoring Trinomials:

• We look for two numbers that multiply to \(3 \times (-2) = -6\) and add to \(1\).
• These numbers are \(3\) and \(-2\).
• Rewrite the middle term: \(3q^2 + 3q - 2q - 2\).
• Factor by grouping: \(3q(q + 1) - 2(q + 1) = (3q - 2)(q + 1)\).

Factor the denominator

Now, we factor the denominator, \(9q^2 - 4\), which is a Difference of Squares:

• Recognize that \(9q^2 = (3q)^2\) and \(4 = 2^2\).
• Applying the difference of squares formula:

$$ 9q^2 - 4 = (3q - 2)(3q + 2) $$

Simplify the rational expression

Substitute the factored forms back into the rational expression to perform Simplifying Rational Expressions:

$$ \frac{(3q - 2)(q + 1)}{(3q - 2)(3q + 2)} $$

We can cancel the common factor \((3q - 2)\) from both the numerator and the denominator, assuming \(3q - 2
eq 0\):

$$ \frac{q + 1}{3q + 2} $$

Answer:

Subtract: \(\frac{3q^2 + q}{9q^2 - 4} - \frac{2}{9q^2 - 4} =\) <blank>\(\frac{q + 1}{3q + 2}\)</blank>