Factor $2x^2 - X - 6$: Step-by-Step Guide

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Have you ever stumbled upon a quadratic expression and felt a little lost trying to factor it? Don't worry, you're not alone! Factoring quadratics can seem tricky at first, but with a clear method and a bit of practice, you'll be a pro in no time. In this article, we'll break down the process of factoring the quadratic expression $2x^2 - x - 6$ step by step. We'll fill in those missing pieces in the equation (2x + oxed{?})(x - oxed{?}) and show you exactly how to arrive at the solution. So, let’s dive in and conquer this factorization challenge together!

Understanding Quadratic Expressions

Before we jump into the specifics, let's quickly recap what a quadratic expression is. A quadratic expression is a polynomial of degree two, which generally takes the form $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our case, the expression $2x^2 - x - 6$ fits this form perfectly, with $a = 2$, $b = -1$, and $c = -6$. Factoring a quadratic expression means rewriting it as a product of two binomials. This is the reverse process of expanding two binomials using the distributive property (also known as the FOIL method).

Why is Factoring Important?

You might be wondering, why bother factoring at all? Well, factoring is a fundamental skill in algebra and has many applications. It's crucial for solving quadratic equations, simplifying algebraic expressions, and even understanding the graphs of quadratic functions (parabolas). When you can factor a quadratic, you can easily find its roots, which are the x-intercepts of the parabola. Factoring also helps in simplifying complex fractions and solving various real-world problems involving quadratic relationships. So, mastering this skill opens doors to more advanced topics in mathematics and its applications.

The General Approach to Factoring

There are several methods to factor a quadratic expression, but one of the most common and effective techniques is the "ac method" or factoring by grouping. This method involves finding two numbers that multiply to 'ac' and add up to 'b'. Once you find these numbers, you can rewrite the middle term (bx) and then factor by grouping. Let's see how this method works in detail as we tackle our specific expression.

Step-by-Step Factoring of $2x^2 - x - 6$

Now, let’s get to the heart of the matter: factoring $2x^2 - x - 6$. We'll follow a systematic approach to ensure we don't miss any steps.

1. Identify a, b, and c

The first step is to identify the coefficients 'a', 'b', and 'c' in our quadratic expression. As we mentioned earlier, for $2x^2 - x - 6$, we have:

• $a = 2$
• $b = -1$
• $c = -6$

These values are crucial for the next steps, so make sure you've identified them correctly.

2. Calculate ac

Next, we need to calculate the product of 'a' and 'c'. This is a key step in the ac method. So,

$ac = 2 imes (-6) = -12$

This number, -12, is what we'll be focusing on in the next step.

3. Find Two Numbers

Now comes the trickiest part: finding two numbers that multiply to 'ac' (-12) and add up to 'b' (-1). This might require a little trial and error, but let’s think systematically. We need two numbers with opposite signs (since their product is negative) and whose difference is 1 (since their sum is -1).

Let's list the factor pairs of 12:

• 1 and 12
• 2 and 6
• 3 and 4

Looking at these pairs, we see that 3 and 4 are the closest in value. If we make 4 negative and 3 positive, we get:

• $3 imes -4 = -12$ (which is our 'ac')
• $3 + (-4) = -1$ (which is our 'b')

So, the two numbers we're looking for are 3 and -4.

4. Rewrite the Middle Term

Now that we've found our numbers, we can rewrite the middle term (-x) using these numbers. Instead of $-x$, we'll write $3x - 4x$. This might seem like a strange move, but it's the key to factoring by grouping. Our expression now looks like this:

$2x^2 + 3x - 4x - 6$

Notice that we haven't changed the value of the expression; we've just rewritten it in a way that makes factoring easier.

5. Factor by Grouping

This is where the “grouping” part of the method comes in. We'll group the first two terms and the last two terms together:

$(2x^2 + 3x) + (-4x - 6)$

Now, we'll factor out the greatest common factor (GCF) from each group. From the first group, $2x^2 + 3x$, the GCF is $x$. Factoring out $x$, we get:

$x(2x + 3)$

From the second group, $-4x - 6$, the GCF is -2. Factoring out -2, we get:

$-2(2x + 3)$

Now, our expression looks like this:

$x(2x + 3) - 2(2x + 3)$

Notice that we have a common binomial factor, $(2x + 3)$, in both terms. This is a good sign – it means we're on the right track!

6. Factor out the Common Binomial

Since $(2x + 3)$ is a common factor, we can factor it out. This is similar to factoring out a single variable, but now we're factoring out an entire expression. Factoring out $(2x + 3)$, we get:

$(2x + 3)(x - 2)$

And there you have it! We've successfully factored the quadratic expression.

7. Fill in the Blanks

Now we can go back to our original question and fill in the blanks:

2x^2 - x - 6 = (2x + oxed{3})(x - oxed{2})

So, the missing numbers are 3 and 2. We've completed the factorization!

Checking Your Answer

It's always a good idea to check your answer to make sure you haven't made any mistakes. The easiest way to do this is to expand the factored form and see if you get back the original expression. Let's expand $(2x + 3)(x - 2)$ using the FOIL method:

• First: $2x imes x = 2x^2$
• Outer: $2x imes -2 = -4x$
• Inner: $3 imes x = 3x$
• Last: $3 imes -2 = -6$

Now, add these terms together:

$2x^2 - 4x + 3x - 6$

Combine like terms:

$2x^2 - x - 6$

This is exactly our original expression, so we know our factorization is correct.

Tips and Tricks for Factoring

Factoring quadratics can become second nature with practice. Here are a few tips and tricks to help you along the way:

• Practice, practice, practice: The more you factor, the better you'll become at recognizing patterns and applying the methods.
• Look for common factors first: Before diving into the ac method, check if there's a common factor you can factor out of the entire expression. This can simplify the problem.
• Use the signs as clues: The signs of 'b' and 'c' can give you clues about the signs of the numbers you're looking for. If 'c' is positive, both numbers have the same sign. If 'c' is negative, the numbers have opposite signs.
• Don't give up: Some quadratics are more challenging than others. If you get stuck, try a different approach or take a break and come back to it later.
• Check your work: Always check your factored form by expanding it to make sure it matches the original expression.

Common Mistakes to Avoid

Even with a solid understanding of the method, it's easy to make mistakes when factoring. Here are some common pitfalls to watch out for:

• Sign errors: Pay close attention to the signs of the numbers, especially when dealing with negative values.
• Incorrectly identifying a, b, and c: Make sure you've correctly identified the coefficients before starting the ac method.
• Forgetting to factor out the GCF: Always look for common factors first to simplify the expression.
• Not checking your answer: Checking your factored form is crucial to catch any errors you might have made.
• Mixing up the factors: Double-check that the numbers you've found multiply to 'ac' and add up to 'b'.

Conclusion

Factoring the quadratic expression $2x^2 - x - 6$ might have seemed daunting at first, but by following our step-by-step guide, you've seen how to break it down into manageable parts. From identifying 'a', 'b', and 'c' to finding the right numbers, rewriting the middle term, and factoring by grouping, you've now equipped yourself with a powerful tool in algebra. Remember, practice makes perfect, so keep working on factoring different types of quadratics, and you'll soon become a factoring master! And remember to always check your answers to ensure accuracy. Happy factoring!

For further learning and practice on factoring quadratic expressions, you can visit websites like Khan Academy's Algebra I section, which offers numerous tutorials and exercises.