Factoring Quadratics: A Deep Dive Into X² - 3x - 28

Hey Plastik Magazine readers! Let's dive into the world of factoring quadratics, specifically focusing on the expression x² - 3x - 28. Factoring is a fundamental skill in algebra, and understanding how to break down these expressions is key to solving equations and grasping more advanced math concepts. This article will break down the process step-by-step, making sure you understand how to arrive at the correct answer.

Understanding the Basics: What is Factoring?

So, what exactly is factoring, anyway? Think of it like this: factoring is the reverse process of multiplying. When you factor an expression, you're essentially breaking it down into a product of simpler expressions (usually binomials, like (x + a) or (x - b)). For example, if we start with the number 12, we can factor it into 3 * 4 or 2 * 6. With quadratic expressions, we want to find two binomials that, when multiplied together, give us the original quadratic. This whole process can seem a bit intimidating at first, but with a bit of practice and by using the right approach, you will become a factoring pro. In our case, the expression is x² - 3x - 28. We want to find two binomials that, when multiplied, give us this expression. The general form of a quadratic equation is ax² + bx + c. Here, a=1, b=-3, and c=-28. Now, let's look at the multiple choices provided to determine which ones are correct. Remember, the goal is to find the expression that is equivalent to x² - 3x - 28.

Methodical Approach: Step-by-Step Factoring

Alright, let's get down to the nitty-gritty and factor the expression x² - 3x - 28. There are a couple of methods we can use, but we'll focus on the most common and straightforward one: factoring by finding two numbers that multiply to 'c' (the constant term) and add up to 'b' (the coefficient of the x term). In our equation, the number 'c' is -28, and the number 'b' is -3. This means we need to find two numbers that multiply to -28 and add up to -3. Let's start listing the factors of -28. We have the following pairs: (1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), (-4, 7). Now, we add each pair to see which one adds up to -3. If we add (1, -28), we get -27. If we add (-1, 28), we get 27. If we add (2, -14), we get -12. If we add (-2, 14), we get 12. Finally, we have (4, -7). Adding them together gives us -3! So, the numbers we are looking for are 4 and -7.

Since we've found our two magic numbers (4 and -7), we can now write our factored expression. Because the coefficient of the x² term is 1, the factored form will be in the format (x + p)(x + q), where p and q are our special numbers. In our case, we get (x + 4)(x - 7). This expression should now be equivalent to the original, x² - 3x - 28. To make sure, we can multiply it back out (foil method). This means you take the first term of the first binomial (x) times the first term of the second binomial (x), this gives us x². Then, you multiply the first term of the first binomial (x) by the second term of the second binomial (-7), and you get -7x. After that, you multiply the second term of the first binomial (4) by the first term of the second binomial (x), which gives you 4x. Finally, multiply the second term of the first binomial (4) by the second term of the second binomial (-7), and you get -28. Therefore, the resulting expression is x² - 7x + 4x - 28. Then, we can combine the like terms and get x² - 3x - 28. As you can see, this is the same expression that we originally had. So, our factored form is correct.

Analyzing the Answer Choices: Finding the Correct Match

Okay, now let's analyze the answer choices provided in the question to match our answer, (x + 4)(x - 7). We already know that our factored expression is (x + 4)(x - 7), but let's carefully go through each of the answer choices to confirm our conclusion and understand why the other options are incorrect.

A. (x - 7)(x + 4): This matches our correctly factored expression! The order of the factors doesn't change the outcome because multiplication is commutative. So, this is the correct answer. The order in which the binomials are written does not change the overall result, so we can consider this the correct one.

B. (x + 7)(x - 4): Let's multiply these binomials to see if they result in x² - 3x - 28. Using the FOIL method, we get x² - 4x + 7x - 28, which simplifies to x² + 3x - 28. This does not match our original expression, so this is incorrect. Notice the sign of the x term is different (+3x instead of -3x), so this is not the right choice. Pay close attention to the signs when factoring, it's easy to make mistakes here!

C. (x + 14)(x - 2): Again, let's multiply these binomials: x² - 2x + 14x - 28, which simplifies to x² + 12x - 28. This doesn't match our original expression either. The middle term is completely different, showing that these factors won't work.

D. (x + 14)(x + 2): Multiplying these: x² + 2x + 14x + 28, which simplifies to x² + 16x + 28. This also isn't a match. This has a positive constant term, which is a major red flag, and the x term is way off. This shows us that this option is completely wrong.

Conclusion: The Final Answer

Therefore, after carefully analyzing the answer choices and comparing them with our factored expression, we can confidently say that A. (x - 7)(x + 4) is the completely factored form of the expression x² - 3x - 28. By following the steps outlined above, you can successfully factor any quadratic expression of this form! Remember, practice makes perfect. The more you factor, the easier and faster it will become. Keep practicing, and you'll become a factoring wizard in no time. If you get stuck, go back to basics, and don't be afraid to try different combinations of factors. Keep up the good work, guys! Happy factoring!