Final answer:
An expression can be factored using the difference of two squares when it is in the form a² - b² and both terms are perfect squares. This identity factors into (a + b)(a - b) and greatly simplifies solving equations that include a difference of squares.
Step-by-step explanation:
An expression can be factored using the difference of two squares when it is in the form a² - b², where both a and b are real numbers, algebraic expressions, or terms that can each be squared. The factoring of the difference of two squares is based on the identity a² - b² = (a + b)(a - b). This means that for any terms a and b which we can square, the expression a² - b² will factor into the product of the sum and difference of a and b. It is important to recognize that this method only applies to subtraction (the difference) and not to addition.
For example, if we have x² - 9, this is a difference of two squares because 9 is a perfect square (3²) and can be factored into (x + 3)(x - 3). When solving equations, recognizing a difference of squares can greatly simplify the process, especially when compared to methods like the quadratic formula or completing the square.