Algebra

(x^2 - 9)/(x - 3) = 5

Step-by-step solution with explanation

Final Answer

x = 8, but note x ≠ 3

Step-by-step solution

1

Factor the numerator completely

x² - 9 is a difference of squares. It factors into (x+3)(x-3). This is a key pattern to recognize.
2

Rewrite the fraction with factored form

We replace the numerator with its factored form. This sets us up to simplify the fraction.
3

Cancel the common factor, note restriction

We cancel (x-3) from the top and bottom. We must note that x ≠ 3, because the original expression is undefined when x = 3 (division by zero).
4

Solve the simple linear equation

Subtract 3 from both sides to isolate x. This gives us our answer.
5

Check the answer is valid

Since x = 2 does not equal the restricted value of 3, our answer is valid. If we had gotten x = 3, there would be no solution.

Understanding this problem

Learning Insight

Canceling (x-3) from the fraction is only valid because it is a common factor — not just a common term. The restriction x ≠ 3 exists because plugging x = 3 into the original equation causes division by zero, which is undefined in math.

Quick Tip

Always write down the restriction BEFORE you cancel. Ask: 'What value makes the denominator zero?' Ban that value from your answer set.

Common Mistake

Students cancel (x-3) and then forget the restriction, accepting x = 3 as a valid answer if it appears. Always check your solution against the original denominator.