Algebra

(x^2 - 4) / (x - 2) = x + 2

Step-by-step solution with explanation

Final Answer

The equation is true for all x ≠ 2 (an identity)

Step-by-step solution

1

Factor the numerator completely

x² - 4 is a difference of squares. The pattern is a² - b² = (a+b)(a-b), so x² - 4 = (x+2)(x-2).
2

Rewrite the left side with factors

Replace the numerator with its factored form. This sets us up to cancel a common factor.
3

Cancel the common factor

The factor (x - 2) appears in both the numerator and denominator, so they cancel out. This leaves just x + 2.
4

State the domain restriction

We cancelled (x - 2), but division by zero is undefined. So x = 2 is not allowed. The equation holds for every other real number.
5

Confirm both sides are equal

After simplifying the left side we get x + 2, which matches the right side exactly. This means the equation is an identity — true for all x except x = 2.

Understanding this problem

Learning Insight

A difference of squares (a² - b²) always factors into (a+b)(a-b). When a factor cancels from a fraction, the result is simpler but you must remember the original denominator could not be zero — that value is excluded from the domain.

Quick Tip

Whenever you see x² minus a perfect square over a linear term, try difference-of-squares factoring first. It almost always creates a cancelable factor.

Common Mistake

Students often say x = 2 is a valid solution after canceling, forgetting that plugging x = 2 into the original expression causes division by zero, which is undefined.