Rectangle Area: Finding Side Lengths With Factoring

by Mei Lin 52 views

Hey guys! Let's dive into a fun math problem that combines geometry and algebra. We're going to explore how to find the side lengths of a rectangle when we know its area is represented by a quadratic expression. This is a super practical application of factoring, and it's something you'll definitely use again in future math courses.

Understanding the Problem

So, here's the scenario: We have a rectangle, and its area is given by the expression x² - 4x - 12 square units. Our mission, should we choose to accept it (and we do!), is to figure out what the side lengths of this rectangle could be. The options we have are:

  • A. (x + 2) and (x - 6)
  • B. (x + 6) and (x - 2)

To solve this, we need to remember a fundamental concept: The area of a rectangle is calculated by multiplying its length and width. In other words:

Area = Length × Width

In our case, the area is the quadratic expression x² - 4x - 12. This means that the side lengths must be two expressions that, when multiplied together, give us this quadratic. This is where factoring comes in handy. Factoring is like the reverse of expanding brackets – we're trying to break down the quadratic into two binomials.

Why Factoring is Key

Factoring is crucial here because it allows us to decompose the area expression into its constituent parts – the length and the width. Think of it like this: if you know the area of a rectangular garden is 24 square feet, you might guess the sides could be 6 feet and 4 feet (since 6 × 4 = 24). Factoring is essentially a systematic way of doing this when the area is given as an algebraic expression. In this particular question, our main goal is to use factoring to rewrite the quadratic expression x² - 4x - 12 as a product of two binomials. These binomials will then represent the possible side lengths of the rectangle.

Now, before we jump into the factoring process itself, let's take a moment to appreciate the beauty and power of algebra. It's not just about abstract symbols and equations; it's a tool that helps us model and solve real-world problems. In this case, we're using algebra to connect the geometric concept of area with the algebraic skill of factoring. This connection is what makes mathematics so fascinating and useful.

The Factoring Process Explained

Now, let's roll up our sleeves and get into the nitty-gritty of factoring the quadratic expression x² - 4x - 12. The general strategy for factoring a quadratic of the form ax² + bx + c (where a = 1 in our case) involves finding two numbers that:

  1. Multiply to give c (the constant term, which is -12 in our case).
  2. Add up to give b (the coefficient of the x term, which is -4 in our case).

So, we're on a quest to find two numbers that multiply to -12 and add up to -4. Let's systematically consider the factors of -12:

  • 1 and -12 (add up to -11)
  • -1 and 12 (add up to 11)
  • 2 and -6 (add up to -4) – Bingo!
  • -2 and 6 (add up to 4)
  • 3 and -4 (add up to -1)
  • -3 and 4 (add up to 1)

As you can see, the pair of numbers that fits our criteria perfectly is 2 and -6. They multiply to -12 and add up to -4. This is the magic combination we need for factoring our quadratic.

Connecting the Numbers to the Factors

Once we've identified the numbers 2 and -6, we can rewrite the quadratic expression in factored form. Remember, the factored form will look like this:

(x + first number) (x + second number)

In our case, this translates to:

(x + 2) (x - 6)

And there you have it! We've successfully factored the quadratic x² - 4x - 12 into (x + 2) (x - 6). This means that the side lengths of the rectangle can be represented by these two binomial expressions. The expressions (x + 2) and (x - 6) represent the length and width of the rectangle. When these two expressions are multiplied together, the result is the quadratic expression x² - 4x - 12, which represents the area of the rectangle. It's a beautiful demonstration of how algebra and geometry intertwine.

Identifying the Correct Option

Now that we've factored the quadratic expression, we can confidently identify the correct option from the choices provided. We found that the factored form of x² - 4x - 12 is (x + 2) (x - 6). Looking back at the options:

  • A. (x + 2) and (x - 6) – This matches our factored form!
  • B. (x + 6) and (x - 2) – This is incorrect.

Therefore, the correct answer is option A. The side lengths that should be used to model the rectangle are (x + 2) and (x - 6).

Why Option B is Incorrect

It's important to understand why option B, (x + 6) and (x - 2), is incorrect. While these binomials look similar to the correct answer, multiplying them together will result in a different quadratic expression. Let's quickly expand (x + 6) (x - 2) to see what we get:

(x + 6) (x - 2) = x² - 2x + 6x - 12 = x² + 4x - 12

Notice that the middle term is +4x, not -4x as in our original area expression. This highlights the importance of getting the signs correct when factoring quadratics. A small change in the signs can lead to a completely different result.

Real-World Applications and Deeper Dive

Now, you might be wondering,