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

Verified step by step guidance

1. Step 1: Recognize that the given expression is a difference of two squares. The difference of two squares is a term that is squared minus another term that is squared. In this case, $x^2$ and $144$ are the two squares.
2. Step 2: The formula for factoring the difference of two squares is $a^2 - b^2 = (a - b)(a + b)$. Here, $a$ is the square root of the first term and $b$ is the square root of the second term.
3. Step 3: In this problem, $a$ is $x$ (since $x^2$ is the first term) and $b$ is $12$ (since the square root of $144$ is $12$).
4. Step 4: Substitute $x$ for $a$ and $12$ for $b$ in the formula. This gives you $(x - 12)(x + 12)$.
5. Step 5: So, the factored form of $x^2 - 144$ is $(x - 12)(x + 12)$.

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 identity is crucial for simplifying expressions where two perfect squares are subtracted, allowing for easier manipulation and solving of equations.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In the context of the expression x^2 - 144, both x^2 and 144 are perfect squares, as 144 is the square of 12. Recognizing perfect squares is essential for applying the difference of squares formula effectively.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In this case, factoring x^2 - 144 involves identifying the two perfect squares and applying the difference of squares formula to express the equation in a more manageable form.