In Exercises 39–48, factor the difference of two squares. x^2−144

Verified step by step guidance

1. Identify the structure of the expression. The given expression is x^2 - 144, which is a difference of two squares. Recall that the difference of two squares has the form a^2 - b^2.
2. Recognize the terms in the expression. Here, x^2 is the square of x, and 144 is the square of 12. So, the expression can be written as (x)^2 - (12)^2.
3. Apply the difference of squares formula: a^2 - b^2 = (a - b)(a + b). In this case, a = x and b = 12.
4. Substitute the values of a and b into the formula. This gives (x - 12)(x + 12).
5. Verify the factorization by expanding (x - 12)(x + 12) to ensure it simplifies back to x^2 - 144. This confirms the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic identity that states that the expression a^2 - b^2 can be factored into (a - b)(a + b). This concept is crucial for simplifying expressions that fit this form, allowing for easier manipulation and solving of equations.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the difference of squares, it involves identifying two square terms and applying the difference of squares formula to rewrite the expression in a more manageable form.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. In the expression x^2 - 144, both x^2 and 144 are perfect squares, with 144 being the square of 12. Recognizing perfect squares is essential for applying the difference of squares factoring technique effectively.