Factoring Quadratics: A Step-by-Step Guide

Hey guys! Let's dive into the world of factoring polynomials, specifically the quadratic expression $x^2-10x+24$. Factoring might seem a bit tricky at first, but trust me, with a little practice, you'll become a pro. This guide will break down the process step by step, making it super easy to understand and apply. We'll go through the question together, exploring the correct method to arrive at the solution. So, grab your pens and paper, and let's get started! Understanding how to factor quadratics is a fundamental skill in algebra, and it opens the door to solving a wide range of problems. Being able to quickly identify the factors of a quadratic expression allows you to find the roots of the equation, which is critical for various applications, from physics to engineering. So, let's get to the core of it, shall we?

Understanding the Problem: $x^2 - 10x + 24$

Alright, before we jump into the solution, let's take a moment to really understand what we're dealing with. The expression $x^2-10x+24$ is a quadratic polynomial. It's in the standard form of $ax^2 + bx + c$, where 'a' is 1 (in our case, since there's no coefficient written in front of $x^2$, it's assumed to be 1), 'b' is -10, and 'c' is 24. Our goal is to rewrite this expression as a product of two binomials (expressions with two terms). In simpler terms, we're looking for two numbers that multiply to give us 'c' (24) and add up to 'b' (-10). This is the core principle behind factoring quadratics, guys! When we factor, we're essentially reversing the process of expanding (or multiplying out) binomials. Remember when you learned about the distributive property and how you'd multiply terms? Factoring is just undoing that, but instead of a single step, we'll be doing multiple operations. The value of understanding how to do this extends far beyond just this single problem, as these skills are highly relevant in other advanced math courses. So, as we solve this problem, remember we are strengthening a skill that could prove very beneficial in your future academic endeavors.

Strategy: Finding the Right Pair

The key to solving this equation is finding the correct pair of numbers. In this case, we need to find two numbers that when multiplied together equal 24 and when added together equal -10. Here's how we can approach this systematically. Let's list the factor pairs of 24: (1, 24), (2, 12), (3, 8), and (4, 6). Now, we need to consider the signs. Since our 'b' term is negative (-10) and our 'c' term is positive (24), we know that both numbers in our pair must be negative. This is because a negative times a negative gives a positive (the 'c' term), and adding two negatives results in a negative (the 'b' term). So, let's change the factor pairs to include negative signs: (-1, -24), (-2, -12), (-3, -8), and (-4, -6). Let's test each of these pairs to see which one adds up to -10. Looking closely, we can see that -4 + (-6) = -10. Bingo! We've found our numbers. Knowing how to recognize the right factors is a super important skill that saves you time, and effort when you do more complicated problems. Understanding this concept now will make future problems easier.

Constructing the Binomials

Now that we have our numbers (-4 and -6), we can write the factored form of the quadratic expression. We'll create two binomials using these numbers: (x - 4) and (x - 6). These are the factors. The variable 'x' comes from the $x^2$ term in the original expression. The numbers -4 and -6 come from our earlier calculations. So, our factored form is (x - 4)(x - 6). Let's double-check by multiplying these binomials back out to make sure we get our original expression. This is super crucial! Multiplying (x - 4)(x - 6) using the FOIL method (First, Outer, Inner, Last), we get: x * x = $x^2$, x * -6 = -6x, -4 * x = -4x, and -4 * -6 = 24. Combining these terms gives us $x^2 - 6x - 4x + 24$, which simplifies to $x^2 - 10x + 24$. As we see, our answer is correct! Keep in mind that this verification step is vital, and can save you from some errors. So, taking an extra minute to double-check is a great habit to develop. Factoring, in general, is an essential skill not just for algebra but for many other mathematical concepts. It helps you simplify expressions, solve equations, and understand the relationships between different parts of an equation.

Choosing the Correct Answer

Now, let's look at the multiple-choice options and select the correct one.

A. (x + 4)(x - 6) B. (x - 4)(x + 6) C. (x + 4)(x + 6) D. (x - 4)(x - 6) E. not factorable

Based on our calculation, the correct factored form is (x - 4)(x - 6). Comparing this to the options, we see that option D. (x - 4)(x - 6) matches our answer perfectly. Thus, D is the correct answer. Make sure you're always careful when choosing the answer, and take your time to make sure you don't make any mistakes. Understanding how to factor is critical to solving a vast array of algebraic problems. It's also great practice for working with more complex equations and it builds your critical thinking and problem-solving abilities. You'll definitely see factoring pop up in your math journey, so it's best to master it now! Taking the time to understand the process and practice it will serve you well in the long run.

Conclusion: Key Takeaways

So, there you have it, guys! We've successfully factored the quadratic expression $x^2 - 10x + 24$. The correct answer is D. (x - 4)(x - 6). Here are the key takeaways from this exercise:

• Identify the coefficients: Understand the values of a, b, and c in the quadratic expression. Remember that 'a' is the coefficient of the $x^2$ term, 'b' is the coefficient of the x term, and 'c' is the constant term. Knowing this will help you quickly identify the terms to work with.
• Find factor pairs: List out the factor pairs of 'c'. Always take your time and carefully list the factors. This is a fundamental step in factoring quadratic equations. Missing a factor pair can lead to incorrect results, so ensure you're thorough.
• Consider the signs: Determine the correct signs (positive or negative) for your factor pairs based on the signs of 'b' and 'c'. Remember that if 'c' is positive, both factors have the same sign. If 'b' is negative, both factors are negative. If 'c' is negative, then one factor is positive, and one is negative.
• Verify your answer: Multiply the binomials you created to make sure they match the original expression. Use the FOIL method to expand and double-check that the result is the original quadratic expression. It's an excellent practice to ensure you've factored correctly. This step is often overlooked, but it is vital to catch any mistakes. Make it a habit to check your work. Remember that practice is key to mastering any skill, and factoring is no exception. Working through different examples will help you become more comfortable with the process, and you'll be able to factor quadratics with speed and confidence. So, keep practicing, and don't be afraid to ask questions if you get stuck!

Keep up the great work, and keep practicing! Math can be fun!