Factorable Expression: Find The Value Of 'a'

Hey guys! Let's dive into a fun math problem today where we're trying to figure out what value of 'a' will make the expression $x^2 - a$ completely factorable. This is a classic algebra question that touches on the difference of squares, a super useful concept in factoring. So, grab your thinking caps, and let’s break it down together!

Understanding the Difference of Squares

Before we jump into the specific problem, let's quickly refresh our understanding of the difference of squares. The difference of squares is a pattern that looks like this: $a^2 - b^2$. It's called the "difference" of squares because we're subtracting one square from another. The cool thing about this pattern is that it factors very neatly into $(a + b)(a - b)$. This is a fundamental concept in algebra, and recognizing this pattern can save you a lot of time and effort when factoring more complex expressions.

So, why is this pattern so important? Well, it allows us to take a seemingly simple expression and break it down into its factors. Factoring, in general, is like reverse multiplication; it's finding the expressions that, when multiplied together, give you the original expression. The difference of squares is a shortcut for factoring expressions that fit this specific pattern. Imagine you have an expression like $x^2 - 9$. You can immediately recognize that 9 is $3^2$, so the expression fits the difference of squares pattern. This means you can factor it into $(x + 3)(x - 3)$ without having to go through a more complicated factoring process.

In our case, we have $x^2 - a$. To make this expression factorable using the difference of squares, 'a' needs to be a perfect square. A perfect square is a number that can be obtained by squaring an integer (a whole number). For example, 4 is a perfect square because it's $2^2$, 9 is a perfect square because it's $3^2$, and so on. This is because when we have $x^2 - a$, we want 'a' to be something we can take the square root of and get a nice, whole number. This whole number will then be the 'b' in our $(x + b)(x - b)$ factored form. Knowing this, we can look at the options provided and quickly narrow down which values of 'a' will work.

Analyzing the Options

Now, let's look at the options provided and see which one fits the bill. We have: A. 12, B. 36, C. 49, and D. 81. Remember, we're looking for a value of 'a' that is a perfect square. This means we need a number that has an integer square root. Let's go through each option:

• A. 12: Is 12 a perfect square? Well, the square root of 12 is somewhere between 3 and 4 (since $3^2 = 9$ and $4^2 = 16$), but it's not a whole number. So, 12 is not a perfect square.
• B. 36: What about 36? The square root of 36 is 6 (since $6^2 = 36$). Bingo! 36 is a perfect square.
• C. 49: Let's check 49. The square root of 49 is 7 (since $7^2 = 49$). Another perfect square! This is great; we're on the right track.
• D. 81: Finally, let's look at 81. The square root of 81 is 9 (since $9^2 = 81$). Yep, 81 is also a perfect square.

So, we've identified that 36, 49, and 81 are all perfect squares. This means that if we substitute any of these values for 'a' in the expression $x^2 - a$, we should be able to factor it using the difference of squares pattern. Let’s see what that looks like.

Factoring with Perfect Squares

Let's take each perfect square we identified and plug it into our expression $x^2 - a$ to see how it factors:

• If a = 36: Our expression becomes $x^2 - 36$. This fits the difference of squares pattern, where $x^2$ is one square and $36$ (which is $6^2$) is the other square. So, we can factor it as $(x + 6)(x - 6)$.
• If a = 49: Our expression becomes $x^2 - 49$. Again, this is a difference of squares, with $x^2$ and $49$ (which is $7^2$). Factoring it gives us $(x + 7)(x - 7)$.
• If a = 81: Our expression is now $x^2 - 81$. This factors into $(x + 9)(x - 9)$ because 81 is $9^2$.

Notice how in each case, because 'a' was a perfect square, we were able to easily factor the expression into two binomials using the difference of squares pattern. This is the key to solving the problem. We were looking for a value of 'a' that would allow us to do this. Now that we've confirmed that 36, 49, and 81 all work, let's go back to the original question and the options to pick the correct answer.

Picking the Correct Answer

Okay, so we know that values of 'a' that are perfect squares will make the expression $x^2 - a$ completely factorable. We identified 36, 49, and 81 as perfect squares from the options given. Looking back at the options: A. 12, B. 36, C. 49, and D. 81, we can see that options B, C, and D are all valid.

However, the question seems to imply there's only one correct answer. This might seem a bit confusing, but it highlights an important point: in math problems, sometimes there might be more than one solution that fits the criteria. In this case, 36, 49, and 81 all make the expression factorable. If this were a multiple-choice question on a test, and only one of these options were available, that would be the correct answer. But since we have three options that work, it's good to recognize that and understand why they all fit the requirements.

In a real-world scenario, this could happen too. For example, if you were designing something where this factoring concept applied, you might have multiple valid solutions depending on other constraints or factors in your design. The important takeaway here is to understand the underlying concept (the difference of squares) and how to apply it to solve the problem.

Final Thoughts

So, to wrap things up, the values of 'a' that make the expression $x^2 - a$ completely factorable are those that are perfect squares. From the given options, 36, 49, and 81 all fit this criterion. This problem is a great illustration of the power of recognizing patterns in math, especially the difference of squares. By understanding this concept, you can quickly and easily factor expressions that might otherwise seem tricky. Keep practicing, guys, and you'll become factoring pros in no time!

Remember, math isn't just about getting the right answer; it's about understanding the process and the logic behind it. When you understand why something works, you can apply that knowledge to solve a wide range of problems. Keep exploring, keep questioning, and keep learning!