Factoring Quadratic Expressions A Step-by-Step Guide To Factoring 10x² + 17x + 3

Factoring quadratic expressions is a fundamental skill in algebra. It's the process of breaking down a quadratic equation into the product of two binomials. This article will guide you through the steps to factor the quadratic expression 10x² + 17x + 3. We'll explore the concepts and methodologies involved, ensuring you have a comprehensive understanding of the topic. This skill is crucial for solving quadratic equations, simplifying algebraic fractions, and tackling more advanced mathematical problems.

Understanding Quadratic Expressions

Before we dive into the factoring process, let's understand what a quadratic expression is. A quadratic expression is a polynomial of degree two, generally represented in the form ax² + bx + c, where a, b, and c are constants, and x is a variable. In our case, the expression 10x² + 17x + 3 fits this form perfectly, with a = 10, b = 17, and c = 3. The goal of factoring is to rewrite this expression as a product of two binomials, such as (px + q)(rx + s). This process involves identifying the correct values for p, q, r, and s that satisfy the original quadratic expression. Factoring is essentially the reverse operation of expanding binomials, a skill that often requires a blend of intuition and systematic methods.

Methods for Factoring

There are several methods for factoring quadratic expressions, but the one we will focus on here is the 'ac' method, sometimes also known as the factoring by grouping method. This method is particularly useful when the coefficient of the x² term (a) is not equal to 1, as is the case in our example. The 'ac' method involves these primary steps:

1. Multiply 'a' and 'c': Calculate the product of the coefficients a and c. In our expression, this would be 10 * 3 = 30. This product is a crucial value that guides the rest of the factoring process.
2. Find two factors of 'ac' that add up to 'b': Look for two numbers that multiply to the value of 'ac' (30 in our case) and add up to the coefficient 'b' (17 in our case). This step is often the trickiest part of factoring, requiring some trial and error and an understanding of number properties. For our expression, the numbers 15 and 2 satisfy these conditions because 15 * 2 = 30 and 15 + 2 = 17.
3. Rewrite the middle term: Rewrite the middle term (bx) using the two factors found in the previous step. In our case, we'll rewrite 17x as 15x + 2x, so our expression becomes 10x² + 15x + 2x + 3. This step transforms the trinomial into a four-term polynomial, setting the stage for factoring by grouping.
4. Factor by grouping: Group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group. This step reveals a common binomial factor that can be extracted from the entire expression. For our expression, we group (10x² + 15x) and (2x + 3). From the first group, we can factor out 5x, leaving us with 5x(2x + 3). The second group (2x + 3) already looks like our target binomial factor. This common binomial factor is the key to the final step.
5. Factor out the common binomial: Factor out the common binomial factor from the two groups. This step brings the expression into its fully factored form. In our example, we factor out (2x + 3) from the expression 5x(2x + 3) + 1(2x + 3), resulting in (2x + 3)(5x + 1). This is the factored form of the original quadratic expression.

Each of these steps plays a vital role in the factoring process. Mastering them allows you to confidently factor a wide range of quadratic expressions.

Step-by-Step Factoring of 10x² + 17x + 3

Let's apply the 'ac' method to factor the expression 10x² + 17x + 3 step-by-step.

Step 1: Multiply 'a' and 'c'

As mentioned earlier, the first step involves multiplying the coefficients a and c. In our expression, a = 10 and c = 3. Therefore, ac = 10 * 3 = 30. This value, 30, is crucial as it guides us in finding the right factors for the next step. It represents the product that the two numbers we are seeking must equal.

Step 2: Find two factors of 'ac' that add up to 'b'

Now, we need to find two numbers that multiply to 30 (the value of ac) and add up to 17 (the value of b). This step often involves some trial and error, but a systematic approach can make it more manageable. We can start by listing pairs of factors of 30:

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

Among these pairs, we are looking for the pair that adds up to 17. By examining the list, we can see that the pair 2 and 15 satisfies this condition: 2 * 15 = 30 and 2 + 15 = 17. These two numbers are the key to rewriting the middle term of our quadratic expression.

Step 3: Rewrite the middle term

Using the factors we found (2 and 15), we rewrite the middle term, 17x, as the sum of 2x and 15x. This transforms our expression 10x² + 17x + 3 into 10x² + 15x + 2x + 3. This step is essential because it breaks the original trinomial into a four-term polynomial, allowing us to use the factoring by grouping method in the next step. By splitting the middle term in this way, we are setting up the expression so that common factors can be extracted, leading to the factored form.

Step 4: Factor by grouping

Now, we group the first two terms and the last two terms of the expression and factor out the greatest common factor (GCF) from each group. Our expression 10x² + 15x + 2x + 3 is grouped as (10x² + 15x) + (2x + 3).

• From the first group (10x² + 15x), the GCF is 5x. Factoring out 5x, we get 5x(2x + 3).
• From the second group (2x + 3), there isn't a common factor other than 1. So, we can write it as 1(2x + 3).

Now, our expression looks like this: 5x(2x + 3) + 1(2x + 3). Notice that we now have a common binomial factor in both terms, which is crucial for the next step.

Step 5: Factor out the common binomial

In the previous step, we obtained the expression 5x(2x + 3) + 1(2x + 3). We can see that the binomial (2x + 3) is a common factor in both terms. Factoring out this common binomial, we get (2x + 3)(5x + 1). This is the factored form of the original quadratic expression.

Therefore, the factored form of 10x² + 17x + 3 is (2x + 3)(5x + 1). We have successfully factored the quadratic expression using the 'ac' method.

Verification

To ensure our factoring is correct, we can expand the factored form (2x + 3)(5x + 1) and check if it equals the original expression 10x² + 17x + 3. Expanding the product of the binomials, we use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):

• First: 2x * 5x = 10x²
• Outer: 2x * 1 = 2x
• Inner: 3 * 5x = 15x
• Last: 3 * 1 = 3

Adding these terms together, we get 10x² + 2x + 15x + 3. Combining the like terms (2x and 15x), we get 10x² + 17x + 3, which is the original expression. This confirms that our factoring is indeed correct. Verification is an essential step in any mathematical problem, especially in factoring, as it ensures accuracy and builds confidence in your solution.

Conclusion

In this article, we successfully factored the quadratic expression 10x² + 17x + 3 using the 'ac' method. We meticulously went through each step, from identifying the coefficients and finding the appropriate factors to rewriting the middle term, factoring by grouping, and finally, verifying our result. Factoring quadratic expressions is a crucial skill in algebra, and mastering it opens doors to solving more complex mathematical problems. The 'ac' method, as demonstrated here, provides a systematic approach to factoring, particularly when the leading coefficient is not 1. By understanding the underlying principles and practicing regularly, you can become proficient in factoring quadratic expressions and confidently apply this skill in various mathematical contexts. Remember to always verify your results to ensure accuracy and deepen your understanding of the process. This skill is not just about finding the right answer; it's about developing a deeper understanding of algebraic manipulation and problem-solving strategies. Keep practicing, and you'll find factoring quadratic expressions becomes second nature. Factoring is a cornerstone of algebra, paving the way for more advanced topics and applications in mathematics and other fields.