Subtract the following and simplify the difference if possible. x 2 − 4 x 2 − x − 6 − x + 2 x 2 − x − 6 \frac{x^2-4}{x^2-x-6}-\frac{x+2}{x^2-x-6}

This problem asks us to subtract the following and simplify the difference if possible. So we have x squared minus four over x squared minus x minus six, and then we have all of that minus x plus two over x squared minus x minus six. Now what I first need to do is look at my two rational expressions that are being subtracted, and see if we have a common denominator. Since I can see that we clearly do, that means I just need to combine the numerators and keep the common denominator. So we'll have x squared minus four, minus x plus two, and all of this is going to be over the common denominator of x squared minus x minus six. Now from here, I can simplify the numerator. Subtracting these quantities is the same thing as having a negative one in front of this first or in front of the second quantity and then adding. It. So in this case, we can keep the x squared minus four the same, then I can just distribute the negative sign into each of these terms. So that'll give me negative x and then negative two, and then we'll this will all be over x squared minus x minus six. So So this is what we get. Now from here I can combine like terms on top. Notice we have a negative four and a negative two, that will combine to give us negative six. So we end up with x squared minus x minus six, and that's all going to be over x squared minus x minus six. And notice something here. We have the same quadratic on top that we do on bottom, the same numerator and the same denominator. We could factor each of these, but we don't actually have to since we can see they're the same thing. So we just know that all of this is going to cancel and reduce down to just giving us one. So one would be the solution to this problem, and that's how you can solve it.

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators

Beginning Algebra

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators

Struggling with Beginning Algebra?

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Multiple Choice

Subtract the following and simplify the difference if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} - \frac{x + 2}{x^{2} - x - 6}$

$\frac{1}{x - 3}$

$x$

$1$

$\frac{x - 3}{x + 2}$

Verified step by step guidance

Identify that both fractions have the same denominator $x^2 - x - 6$, so you can combine the numerators directly by subtraction over the common denominator.

Write the expression as a single fraction: $\frac{x^2 - 4 - (x + 2)}{x^2 - x - 6}$.

Simplify the numerator by distributing the negative sign and combining like terms: $x^2 - 4 - x - 2$.

Factor the denominator $x^2 - x - 6$ by finding two numbers that multiply to $-6$ and add to $-1$, then write it as $(x - 3)(x + 2)$.

Factor the numerator if possible, then simplify the fraction by canceling any common factors between numerator and denominator.

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Struggling with Beginning Algebra?

Subtract the following and simplify the difference if possible.x2−4x2−x−6−x+2x2−x−6\(\frac{x^2-4}{x^2-x-6}\)-\(\frac{x+2}{x^2-x-6}\)

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Subtract the following and simplify the difference if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} - \frac{x + 2}{x^{2} - x - 6}$