Factoring $6x^2 - 17x + 7$: A Step-by-Step Guide

Let's dive into factoring the quadratic expression $6x^2 - 17x + 7$. Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this guide, we'll break down the process step by step, making it easy to understand and apply. This method involves finding two binomials that, when multiplied together, give us the original quadratic expression. It might seem daunting at first, but with a systematic approach, you’ll be factoring like a pro in no time! This method relies on identifying the coefficients and constants in the quadratic and using them to find the correct binomial factors.

Understanding Quadratic Expressions

Before we jump into factoring, let's quickly review what a quadratic expression is. A quadratic expression is a polynomial of degree two, generally written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ is not zero. In our case, the expression is $6x^2 - 17x + 7$, where $a = 6$, $b = -17$, and $c = 7$. Understanding these coefficients is crucial for the factoring process. The goal of factoring is to rewrite this expression as a product of two binomials, like $(px + q)(rx + s)$, where $p$, $q$, $r$, and $s$ are constants. Recognizing the structure of a quadratic expression is the first step towards successfully factoring it.

Step 1: The AC Method

The AC method is a popular technique for factoring quadratic expressions. It involves multiplying the coefficient of the $x^2$ term ($a$) by the constant term ($c$). In our expression, $a = 6$ and $c = 7$, so we multiply them: $6 imes 7 = 42$. The next step is to find two numbers that multiply to this result (42) and add up to the coefficient of the $x$ term ($b$), which is -17. This step is crucial as it sets the foundation for breaking down the middle term. Think of it like finding the right puzzle pieces that fit perfectly together. The numbers we're looking for will help us rewrite the quadratic expression in a way that allows us to factor by grouping.

Step 2: Finding the Right Numbers

Now, we need to find two numbers that multiply to 42 and add up to -17. This might involve a bit of trial and error, but a systematic approach will make it easier. Let's list the factor pairs of 42: (1, 42), (2, 21), (3, 14), and (6, 7). Since we need the numbers to add up to -17, we'll consider the negative pairs: (-1, -42), (-2, -21), (-3, -14), and (-6, -7). Looking at these pairs, we see that -3 and -14 fit the bill: $-3 imes -14 = 42$ and $-3 + (-14) = -17$. These numbers are key to rewriting our quadratic expression. Identifying the correct pair of numbers is the most critical step in this method. These numbers will allow us to split the middle term and proceed with factoring by grouping.

Step 3: Rewriting the Middle Term

Using the numbers -3 and -14, we rewrite the middle term (-17x) of our quadratic expression. Instead of $-17x$, we write $-3x - 14x$. So, our expression $6x^2 - 17x + 7$ becomes $6x^2 - 3x - 14x + 7$. This step is crucial because it sets up the expression for factoring by grouping. By splitting the middle term, we create two pairs of terms that share common factors. This allows us to simplify the expression and ultimately factor it into two binomials. Rewriting the middle term is a clever algebraic trick that makes the factoring process more manageable.

Step 4: Factoring by Grouping

Now, we factor by grouping. We group the first two terms and the last two terms: $(6x^2 - 3x) + (-14x + 7)$. Next, we find the greatest common factor (GCF) in each group. In the first group, the GCF is $3x$, and in the second group, the GCF is -7. Factoring out the GCF from each group, we get: $3x(2x - 1) - 7(2x - 1)$. Notice that both terms now have a common binomial factor: $(2x - 1)$. This is a good sign, as it means we're on the right track. Factoring by grouping is a powerful technique that allows us to simplify complex expressions. By identifying common factors, we can rewrite the expression in a more manageable form.

Step 5: Final Factorization

Since both terms have the common factor $(2x - 1)$, we can factor it out: $(2x - 1)(3x - 7)$. And that's it! We have factored the quadratic expression $6x^2 - 17x + 7$ completely. The factored form is $(2x - 1)(3x - 7)$. To verify our result, we can multiply the two binomials back together to see if we get the original expression. This step is a good practice to ensure that the factoring was done correctly. The final factorization represents the quadratic expression as a product of two linear binomials, which is the ultimate goal of factoring.

Verification

Let's verify our factorization by multiplying $(2x - 1)$ and $(3x - 7)$:

$(2x - 1)(3x - 7) = 2x(3x) + 2x(-7) - 1(3x) - 1(-7)$

$= 6x^2 - 14x - 3x + 7$

$= 6x^2 - 17x + 7$

This matches our original expression, so our factorization is correct. Verifying the solution is a crucial step in any mathematical problem. It ensures that the answer is accurate and that no mistakes were made during the process. In this case, multiplying the factors back together confirms that our factorization is correct.

Conclusion

Factoring the quadratic expression $6x^2 - 17x + 7$ involves a systematic approach using the AC method and factoring by grouping. By following these steps, you can confidently factor various quadratic expressions. Remember, practice makes perfect, so keep working on different examples to master this skill. Factoring quadratic expressions is a fundamental skill in algebra that has numerous applications in mathematics and other fields. By mastering this skill, you'll be better equipped to tackle more complex problems. Understanding the process and practicing regularly will help you become proficient in factoring.

For further learning and practice on factoring quadratic expressions, you might find helpful resources on websites like Khan Academy's Algebra I section.