Evaluate the rational expression below at x = 2 x=2 x 2 − 4 x 2 − x − 6 \frac{x^2-4}{x^2-x-6}

This problem asks us to evaluate the rational expression below at x equals two. So, all this means is that we need to take every place we see x and replace it with two. So, that's going to give us two squared minus four, divided by, or I should say over, and that's gonna be two squared, minus two, minus six. So this is the rational expression we end up with when we go ahead and plug in two. Now two squared is four, so we get four minus four over, and then we're going to have two squared, which is four minus two minus six. Now, I could go ahead and keep simplifying from here, but notice something really interesting that happens. When I do this, I end up with four minus four on top, which gives me a zero. So we're going to end up with a zero on top, but what happens if I simplify the bottom? We get four minus two, which is negative two, minus six. Now negative two minus six is negative eight, but that doesn't even matter because we're dividing that into zero. So the whole thing is just gonna come out to zero anyways. So zero would be the solution to this problem. So this is how you can deal with rational expressions when you need to evaluate them at a certain x value like we saw here. Hope you found this helpful.

8. Rational Expressions and Equations

Simplifying Rational Expressions

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Simplifying Rational Expressions

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Multiple Choice

Evaluate the rational expression below at $x equals 2$
$\frac{x^{2} - 4}{x^{2} - x - 6}$

$\infty$

$0$

$\frac12$

$\frac23$

Verified step by step guidance

Start with the given rational expression: $\frac{x^2 - 4}{x^2 - x - 6}$.

Factor both the numerator and the denominator. The numerator $x^2 - 4$ is a difference of squares, so it factors as $(x - 2)(x + 2)$. The denominator $x^2 - x - 6$ factors into two binomials that multiply to $-6$ and add to $-1$, which are $(x - 3)(x + 2)$.

Rewrite the expression using the factored forms: $\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)}$.

Cancel the common factor $(x + 2)$ from numerator and denominator, assuming $x \neq -2$ to avoid division by zero. The simplified expression is then $\frac{x - 2}{x - 3}$.

Evaluate the simplified expression at $x = 2$ by substituting 2 into the expression: $\frac{2 - 2}{2 - 3}$.

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Struggling with Beginning & Intermediate Algebra?

Evaluate the rational expression below at x=2x=2 x2−4x2−x−6\frac{x^2-4}{x^2-x-6}

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Evaluate the rational expression below at $x equals 2$
$\frac{x^{2} - 4}{x^{2} - x - 6}$