Algebra

(x2−9)/(x2−x−6)

Step-by-step solution with explanation

Final Answer

(x + 3)/(x + 2), where x ≠ 3 and x ≠ −2

Step-by-step solution

1

Factor the numerator using difference of squares

The numerator is a difference of two perfect squares: and . We can always factor as , so becomes .
2

Factor the denominator into two binomials

We need two numbers that multiply to −6 and add to −1. Those numbers are −3 and +2, giving us .
3

Rewrite the fraction with factored forms

Replace both the top and bottom with their factored forms. This lets us see what parts they have in common.
4

Cancel the common factor from both parts

Both the top and bottom have , so we can divide it out. This only works when because we can't divide by zero.
5

State the restrictions on the variable

and
The original denominator equals zero when or , which would make the fraction undefined. We must exclude these values even after simplifying.

Understanding this problem

Learning Insight

Simplifying rational expressions works just like reducing fractions with numbers. We factor both the top and bottom, then cancel matching pieces. The key is that we can only cancel factors (things being multiplied), never terms (things being added or subtracted). Always remember to state which x-values make the original denominator zero, because those values are forever off-limits.

Quick Tip

Before you start factoring, quickly check if the numerator is a difference of squares (). If it is, you've got an instant factorization: . This saves time and reduces errors.

Common Mistake

Students often forget to state the restrictions after canceling. Even though cancels out, still makes the original problem undefined, so you must write in your final answer. Never erase the history of what made the denominator zero!