Difference of Two Squares
Factor expressions in the form a² - b² = (a+b)(a-b) by recognizing perfect squares and applying the difference of two squares pattern in Grade 9 Algebra.
Key Concepts
Property The factored form of a difference of two squares is: $a^2 b^2 = (a+b)(a b)$.
Explanation This is the easiest special product to spot! When you see a perfect square minus another perfect square, just take the square root of each. Your factors are simply the sum of those roots and the difference of those roots. No tricky middle term to worry about, just pure subtraction magic!
Examples $4x^2 25 = (2x)^2 5^2 = (2x+5)(2x 5)$ $9m^4 16n^6 = (3m^2)^2 (4n^3)^2 = (3m^2 + 4n^3)(3m^2 4n^3)$ $ 81 + x^{10} = x^{10} 81 = (x^5)^2 9^2 = (x^5+9)(x^5 9)$.
Common Questions
What is the difference of two squares factoring pattern?
The difference of two squares formula is a² - b² = (a + b)(a - b). It applies when you have a perfect square minus another perfect square with no middle term. Take the square root of each term, then write one binomial with a plus and one with a minus sign.
How do you recognize a difference of two squares?
Look for an expression with exactly two terms separated by a subtraction sign where both terms are perfect squares. Examples include x² - 9 (since 9 = 3²) and 25y² - 16 (since 25y² = (5y)² and 16 = 4²). There must be no middle term.
Can the sum of two squares be factored the same way?
No. The sum of two squares a² + b² cannot be factored over real numbers. Only the difference (subtraction) of two perfect squares factors into conjugate binomials (a + b)(a - b). The sum of two squares is considered a prime polynomial in real number algebra.