Factoring Difference of Two Squares

The expression x² – 25 is a difference of squares, which can be factored using the identity a² – b² = (a + b)(a – b). Here, a is x and b is 5, since 25 is 5². Applying this identity, we rewrite the expression as (x + 5)(x – 5). This shows that the original quadratic can be expressed as the product of two binomials, revealing its roots at x = 5 and x = -5.

To factor the expression 4x² – 9, we recognize it as a difference of squares, which follows the formula a² - b² = (a - b)(a + b). Here, a is 2x and b is 3, since (2x)² = 4x² and 3² = 9. Applying the formula, we get (2x - 3)(2x + 3) as the factored form. This shows that the expression can be expressed as the product of two binomials.

The expression 16h² – 49k² is a difference of squares, which can be factored using the formula a² – b² = (a + b)(a – b). Here, 16h² is (4h)² and 49k² is (7k)². Applying the formula, we identify a as 4h and b as 7k. Thus, the expression factors into (4h + 7k)(4h - 7k), representing the two binomials that, when multiplied, yield the original expression.

The expression 64 – b² is a difference of squares, which can be factored using the formula a² - b² = (a + b)(a - b). Here, 64 is 8² and b² is b², so we can apply this formula with a = 8 and b = b. Thus, 64 – b² factors into (8 + b)(8 – b), representing the two binomials that multiply to give the original expression.

To factor the expression x⁴ – 4, we recognize it as a difference of squares, which can be expressed as (x²)² – (2)². This can be factored using the identity a² – b² = (a + b)(a - b). Applying this identity, we have (x² + 2)(x² - 2). The term (x² - 2) can be further factored, but the question specifically asks for the initial factorization, making (x² + 2)(x² - 2) the correct answer.

The expression 25x⁴ – 9y² is a difference of squares, which can be factored using the formula a² - b² = (a + b)(a - b). Here, a is 5x² and b is 3y. Applying this formula, we get (5x² + 3y)(5x² - 3y) as the factored form. This approach effectively simplifies the expression into two binomial factors, showcasing the relationship between the terms and their squares.

The expression 100 – x⁴ can be recognized as a difference of squares. It can be rewritten as (10²) – (x²)², which factors into (10 + x²)(10 – x²). This shows that the expression is decomposed into two binomials, one adding and one subtracting the square terms. The factorization effectively simplifies the original expression, allowing for easier manipulation in algebraic contexts.

To factor the expression 121x² – 144, we recognize it as a difference of squares, which follows the formula a² - b² = (a + b)(a - b). Here, 121x² is (11x)² and 144 is 12². Applying the formula, we set a = 11x and b = 12, leading to (11x + 12)(11x - 12) as the factored form. This method efficiently simplifies the expression into two binomials.