Factoring Quadratics: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of factoring quadratic expressions. It's a super useful skill for simplifying equations and solving problems. We're going to break down how to factor an expression like $48 g^2-22 g h-15 h^2$ and find the correct answer from the multiple-choice options. Let's get started, shall we? This topic is often a stumbling block for many, but fear not! With a little patience and the right approach, you'll be factoring like a pro in no time. This is especially true when dealing with expressions involving two variables, like this one with both 'g' and 'h'. The goal is to rewrite the expression as a product of two binomials. This skill is fundamental in algebra and pops up in tons of different areas, from solving equations to graphing parabolas. The process involves breaking down the quadratic expression into its constituent parts, making it easier to work with and manipulate. We will look at techniques to identify potential factors and then check our work to ensure the expression is correctly factored.

Understanding the Basics of Factoring

Before we jump into our example, let's quickly recap what factoring actually means. Factoring is essentially the reverse process of multiplying. When you multiply two binomials (expressions with two terms), you're expanding them. Factoring is about taking a quadratic expression and rewriting it as a product of two binomials. Think of it like this: you're breaking down a complex shape into its simpler building blocks. In the context of our problem, we want to find two binomials that, when multiplied together, give us the original quadratic expression $48 g^2-22 g h-15 h^2$. This means finding two expressions in the form of (ax + by) and (cx + dy) that multiply to give the original quadratic. This might sound a bit abstract, but we'll see it in action soon. It's a key skill for solving equations, simplifying expressions, and understanding the structure of quadratic functions. Being able to factor is like having a secret code to unlock the secrets of algebra! Plus, it helps simplify problems, making them easier to solve and understand. So, the first step is always identifying the coefficients and constant terms of the expression. This will help guide the factorization process. We need to find the correct combinations of numbers and variables that will eventually lead to the original expression. Remember, practice makes perfect! The more you factor, the easier it gets to spot the correct factors.

Breaking Down the Quadratic Expression

Now, let's get our hands dirty with the expression: $48 g^2-22 g h-15 h^2$. The first step is to analyze the coefficients and the signs. We have 48 as the coefficient of $g^2$, -15 as the constant term (considering $h^2$), and -22 as the coefficient of the middle term (gh). It's important to keep track of the signs (+ or -) because they play a crucial role in determining the signs within the binomials. Our target is to rewrite this as a product of two binomials: ( _g _h ) ( _g _h ). Let’s focus on the first term, $48g^2$. We need to find two factors that multiply to give us 48. These could be 1 and 48, 2 and 24, 3 and 16, 4 and 12, or 6 and 8. The trick is to try different combinations to see which ones work when we consider the other terms in the expression. The signs of the constant and middle terms give clues about the signs within the binomials. Since the constant term (-15) is negative, one binomial will have a positive sign, and the other will have a negative sign. This is because a positive times a negative equals a negative. The middle term (-22gh) is also negative, which helps us to narrow down the possible factors. We need to ensure that when we combine the cross-multiplied terms (those that result from the outer and inner products), we get -22gh. Now, let's focus on -15h². We need to consider the factors that multiply to give us -15. They could be 1 and -15, or -1 and 15, or 3 and -5, or -3 and 5. This is where a bit of trial and error comes in. It helps to write down possible factor pairs for both the coefficient of g² and the constant term to have a clear reference. It makes the process much more organized.

Finding the Correct Factors

Based on the analysis, we have to start testing. Let's consider option A: $(6 g-5 h)(8 g+3 h)$. We'll multiply these binomials using the FOIL method (First, Outer, Inner, Last):

• First: $(6g * 8g) = 48g^2$
• Outer: $(6g * 3h) = 18gh$
• Inner: $(-5h * 8g) = -40gh$
• Last: $(-5h * 3h) = -15h^2$

Now, let's combine the like terms: $48g^2 + 18gh - 40gh - 15h^2 = 48g^2 - 22gh - 15h^2$. Hey, it matches our original expression! We've found the correct factors. This confirms that the correct answer is $(6 g-5 h)(8 g+3 h)$. Sometimes, the only way to find the correct factors is by checking all the options. Therefore, even though we know the answer, it’s worth quickly checking the other options to make sure we've found the only correct one. Doing this helps to reinforce the process and highlights the importance of each step. Let's briefly look at option B: $(6 g+5 h)(8 g-3 h)$. Doing the same FOIL process, we get: $(6g * 8g) = 48g^2$, $(6g * -3h) = -18gh$, $(5h * 8g) = 40gh$, and $(5h * -3h) = -15h^2$. Combining the middle terms, we get: $-18gh + 40gh = 22gh$. Since this doesn't match the middle term of our original expression (-22gh), we know this isn't the correct answer. The difference in sign shows us why this combination does not work. This exercise of trying all of the options makes you extremely familiar with the process of factoring. For the remaining options, we can see that they won’t work because they either have incorrect signs, incorrect coefficients, or mix up the variables. This also helps you to know exactly how to correctly factor a given quadratic expression.

Conclusion: The Correct Answer

So, the answer is A. $(6 g-5 h)(8 g+3 h)$. Congratulations! You've successfully factored the expression $48 g^2-22 g h-15 h^2$. Factoring might seem tricky at first, but with practice, you'll become more confident in identifying the factors. The key is to break down the problem step-by-step, pay close attention to the signs, and don't be afraid to try different combinations. Remember, practice makes perfect! Keep working on similar problems, and you'll become a factoring master in no time. Each quadratic expression can be factored by using the same basic principles: understanding the relationship between the coefficients and the constant term, and then breaking down the expression into simpler binomials. Factoring is a fundamental skill in algebra, which is used in many more complex concepts such as solving equations, simplifying rational expressions, and analyzing functions. Keep practicing, and good luck with your math journey!