Question

what is the value of the rational expression \\(\frac{x^2 - 2x - 3}{4x^2 - 12}\\) when \\(x = 3\\)? \\(\bigcirc -\frac{1}{4}\\) \\(\bigcirc 0\\) \\(\bigcirc \frac{1}{4}\\) \\(\bigcirc\\) undefined

Explanation:

Step1: Substitute \( x = 3 \) into the denominator

The denominator of the rational expression is \( 4x^2 - 12 \). Substitute \( x = 3 \):
\( 4(3)^2 - 12 = 4\times9 - 12 = 36 - 12 = 24 \)? Wait, no, wait. Wait, let's recalculate. Wait, \( 4x^2 - 12 \) when \( x = 3 \): \( 4(3)^2 - 12 = 49 - 12 = 36 - 12 = 24 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, no, wait the original expression is \( \frac{x^2 - 2x - 3}{4x^2 - 12} \). Wait, let's check the denominator again. Wait, \( 4x^2 - 12 = 4(x^2 - 3) \). When \( x = 3 \), \( x^2 - 3 = 9 - 3 = 6 \), so \( 4*6 = 24 \). Wait, but let's check the numerator. Numerator: \( x^2 - 2x - 3 \) when \( x = 3 \): \( 9 - 6 - 3 = 0 \). Wait, numerator is 0? Wait, no, wait \( 3^2 - 2*3 - 3 = 9 - 6 - 3 = 0 \). So the numerator is 0, denominator is 24? Wait, that would be 0. But wait, maybe I made a mistake. Wait, no, let's re-express the rational expression. Let's factor the numerator and denominator.

Numerator: \( x^2 - 2x - 3 \). Let's factor it. We need two numbers that multiply to -3 and add to -2. Those numbers are -3 and +1. So \( (x - 3)(x + 1) \).

Denominator: \( 4x^2 - 12 = 4(x^2 - 3) = 4(x - \sqrt{3})(x + \sqrt{3}) \). Wait, no, \( x^2 - 3 \) factors as \( (x - \sqrt{3})(x + \sqrt{3}) \), but when \( x = 3 \), denominator is \( 4(9 - 3) = 46 = 24 \), numerator is \( (3 - 3)(3 + 1) = 04 = 0 \). So the expression becomes \( \frac{0}{24} = 0 \). Wait, but let's check again. Wait, maybe I made a mistake in factoring. Wait, numerator: \( x^2 - 2x - 3 \). Let's check with \( x = 3 \): \( 9 - 6 - 3 = 0 \). Denominator: \( 4(3)^2 - 12 = 36 - 12 = 24 \). So \( 0/24 = 0 \). So the value is 0.

Wait, but let's confirm. So when we substitute \( x = 3 \) into the rational expression, numerator is 0, denominator is 24, so the value is 0.

Step2: Confirm the calculation

Numerator: \( 3^2 - 2*3 - 3 = 9 - 6 - 3 = 0 \)

Denominator: \( 4*3^2 - 12 = 4*9 - 12 = 36 - 12 = 24 \)

So the expression is \( \frac{0}{24} = 0 \)

Answer:

0