Factoring Quadratics: A Step-by-Step Guide

Let's dive into factoring the quadratic expression $2x^2 + 13x + 15$. Factoring quadratic expressions is a fundamental skill in algebra, and it's super useful for solving equations, simplifying expressions, and even tackling more advanced math problems. In this guide, we'll break down the process step-by-step, so you'll be factoring like a pro in no time! The correct answer is D. $(2x + 5)(x + 3)$. Let's explore how we arrive at this solution.

Understanding Quadratic Expressions

First things first, let's make sure we're all on the same page. A quadratic expression is basically a polynomial with the highest power of the variable being 2. The general form looks like this: $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, we have $2x^2 + 13x + 15$, so $a = 2$, $b = 13$, and $c = 15$. Factoring a quadratic expression means we want to rewrite it as a product of two binomials (expressions with two terms). Think of it like reverse-distributing!

The Factoring Process: A Step-by-Step Approach

Now, let's get down to business and factor $2x^2 + 13x + 15$. There are several methods you can use, but we'll focus on the most common one: the 'ac method'. This method involves a bit of trial and error, but once you get the hang of it, it's super effective.

Step 1: Multiply 'a' and 'c'

The first step is to multiply the coefficient of the $x^2$ term ($a$) by the constant term ($c$). In our case, that's $2 * 15 = 30$. This number, 30, is going to be our guide throughout the factoring process.

Step 2: Find Factors of 'ac' that Add Up to 'b'

Next, we need to find two factors of 30 that, when added together, give us the coefficient of the $x$ term ($b$), which is 13. Let's list out the factor pairs of 30:

• 1 and 30
• 2 and 15
• 3 and 10
• 5 and 6

Looking at these pairs, we can see that 3 and 10 add up to 13. Bingo! These are the numbers we'll use in the next step. This is the crucial step where you need to play detective and find the right combination. Sometimes it might take a bit of trial and error, but don't worry, you'll get there!

Step 3: Rewrite the Middle Term

Now comes the clever part. We're going to rewrite the middle term ($13x$) using the factors we just found (3 and 10). So, we'll replace $13x$ with $3x + 10x$. Our expression now looks like this: $2x^2 + 3x + 10x + 15$. Notice that we haven't actually changed the expression; we've just rewritten it in a way that will help us factor.

Step 4: Factor by Grouping

This is where the magic happens! We're going to group the first two terms and the last two terms together and factor out the greatest common factor (GCF) from each group. Let's break it down:

• From the first group, $2x^2 + 3x$, the GCF is $x$. Factoring out $x$, we get $x(2x + 3)$.
• From the second group, $10x + 15$, the GCF is 5. Factoring out 5, we get $5(2x + 3)$.

Now our expression looks like this: $x(2x + 3) + 5(2x + 3)$. Notice anything special? Both terms have a common factor of $(2x + 3)$. This is exactly what we want!

Step 5: Factor Out the Common Binomial

Since both terms have $(2x + 3)$ as a factor, we can factor it out. This gives us: $(2x + 3)(x + 5)$. And that's it! We've factored the quadratic expression.

Checking Your Answer

It's always a good idea to check your answer, especially in math. To do this, we can simply multiply the two binomials we found using the FOIL method (First, Outer, Inner, Last). Let's see if we get back our original expression:

• First: $2x * x = 2x^2$
• Outer: $2x * 5 = 10x$
• Inner: $3 * x = 3x$
• Last: $3 * 5 = 15$

Adding these together, we get $2x^2 + 10x + 3x + 15$, which simplifies to $2x^2 + 13x + 15$. Hooray! Our factoring is correct.

Common Mistakes to Avoid

Factoring quadratics can be tricky, and there are a few common mistakes that students often make. Keep an eye out for these to avoid unnecessary errors:

• Forgetting to multiply 'a' and 'c': This is a crucial first step, so don't skip it! If you forget, you'll likely end up with the wrong factors.
• Incorrectly identifying factors: Make sure you find the correct factors that add up to 'b'. It's easy to make a mistake here, so double-check your work.
• Not factoring out the GCF: Always look for a greatest common factor before you start the factoring process. This can simplify the expression and make it easier to factor.
• Sign errors: Pay close attention to the signs of the factors and make sure they're correct. A small sign error can throw off your entire answer.

Alternative Methods for Factoring Quadratics

While the 'ac method' is a popular and versatile approach, there are other methods you can use to factor quadratic expressions. Here are a couple of alternatives:

Trial and Error

This method involves making educated guesses for the factors and checking if they work. It can be faster for simpler quadratics, but it can also be more time-consuming for more complex ones. Basically, you try different combinations of binomials until you find the right one. It's like a puzzle – you keep trying pieces until they fit.

The Quadratic Formula

If you're struggling to factor a quadratic expression, the quadratic formula can be a lifesaver. It's a surefire way to find the roots of the quadratic equation, which can then be used to factor the expression. The quadratic formula is: x = rac{-b {±} {\sqrt{b^2 - 4ac}}}{2a}. Once you find the roots, say $r_1$ and $r_2$, you can write the factored form as $a(x - r_1)(x - r_2)$.

Tips and Tricks for Mastering Factoring

Factoring quadratics is a skill that gets easier with practice. Here are a few tips and tricks to help you master it:

• Practice, practice, practice: The more you practice, the better you'll become at recognizing patterns and finding factors quickly.
• Use online resources: There are tons of websites and videos that can help you learn and practice factoring. Khan Academy, for example, has excellent resources on this topic.
• Work with a study group: Studying with friends can make learning more fun and help you understand concepts better.
• Don't be afraid to ask for help: If you're stuck, don't hesitate to ask your teacher, tutor, or classmates for help.

Conclusion

Factoring quadratic expressions might seem daunting at first, but with a clear understanding of the process and plenty of practice, you'll be able to tackle even the trickiest problems. Remember, the key is to break down the problem into smaller steps and take your time. So, keep practicing, and you'll be factoring quadratics like a pro in no time! And remember, the correct factored form of $2x^2 + 13x + 15$ is indeed $(2x + 3)(x + 5)$, option B from our initial choices.

For further learning and practice, you can explore resources like Khan Academy's Algebra 1 section.