Factor $x^2 - A$: Find The Right Value Of $a$

When it comes to simplifying algebraic expressions, factoring is a superpower. It allows us to break down complex polynomials into simpler, multiplied terms. Today, we're diving into a specific type of expression: $x^2 - a$. Our mission, should we choose to accept it, is to figure out which value of $a$ will make this expression completely factored. This isn't just a math puzzle; understanding this concept is fundamental to solving a wide range of algebraic problems, from solving quadratic equations to simplifying rational expressions. Let's unravel the mystery behind making $x^2 - a$ factorable, and we'll explore the options provided to pinpoint the correct answer. We're looking for a value of '$a$' that transforms $x^2 - a$ into a form that can be easily broken down into two binomials. The key to this lies in recognizing a special pattern in algebra: the difference of squares. This pattern, which states that $A^2 - B^2 = (A - B)(A + B)$, is our guiding star. For $x^2 - a$ to fit this pattern, '$x^2$' is already in the form of $A^2$ (where $A=x$), so we need '$a$' to be in the form of $B^2$. In simpler terms, '$a$' must be a perfect square. A perfect square is a number that can be obtained by squaring an integer (multiplying an integer by itself). For example, 4 is a perfect square because $2^2 = 4$, and 9 is a perfect square because $3^2 = 9$. When '$a$' is a perfect square, say $k^2$, then our expression $x^2 - a$ becomes $x^2 - k^2$, which perfectly fits the difference of squares pattern and can be factored into $(x - k)(x + k)$. Now, let's examine the given options to see which one is a perfect square and therefore will make our expression completely factored.

Unpacking the Options: Which $a$ is a Perfect Square?

We're presented with four potential values for $a$: A. 49, B. 12, C. 81, and D. 36. Our goal is to identify the one that is a perfect square, allowing us to factor $x^2 - a$ using the difference of squares formula. Let's take a close look at each option. First, consider Option A: 49. Is 49 a perfect square? Yes, it is! We know that $7 \times 7 = 49$, which means $7^2 = 49$. Therefore, if $a = 49$, our expression becomes $x^2 - 49$, which can be factored as $(x - 7)(x + 7)$. This is a completely factored form, satisfying our condition. Now, let's move on to Option B: 12. Is 12 a perfect square? To check, we can think of integers whose squares are close to 12. We have $3^2 = 9$ and $4^2 = 16$. Since 12 falls between 9 and 16, and there's no integer that, when multiplied by itself, equals 12, 12 is not a perfect square. If $a=12$, the expression $x^2 - 12$ cannot be factored into simple binomials with integer coefficients using the difference of squares method. Next, let's evaluate Option C: 81. Is 81 a perfect square? Absolutely! We know that $9 \times 9 = 81$, so $9^2 = 81$. If $a = 81$, the expression $x^2 - 81$ can be factored into $(x - 9)(x + 9)$. This is also a completely factored form. Finally, we examine Option D: 36. Is 36 a perfect square? Yes, indeed! $6 \times 6 = 36$, which means $6^2 = 36$. Thus, if $a = 36$, the expression $x^2 - 36$ can be factored as $(x - 6)(x + 6)$. This, too, is a completely factored form. We've identified three options (49, 81, and 36) that are perfect squares and would allow $x^2 - a$ to be completely factored. However, in a multiple-choice question, there's typically only one correct answer. Let's re-read the question carefully: "Which value of $a$ would make the following expression completely factored?" This implies we need to select one value from the given options that fulfills the condition. All three perfect squares (49, 81, and 36) will indeed make the expression completely factored. This suggests there might be an implicit assumption or a standard convention at play, or perhaps the question intends for us to select any value that works. In typical mathematical assessment contexts, if multiple options satisfy a condition, there might be an error in the question design or it's a