Factoring The Quadratic Expression 5p^2 - P - 18 A Step-by-Step Guide

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Factoring quadratic expressions is a fundamental skill in algebra. It allows us to rewrite complex expressions into simpler forms, making them easier to solve, analyze, and manipulate. In this comprehensive guide, we will walk you through the process of factoring the quadratic expression 5p^2 - p - 18. We will break down the steps involved, explain the reasoning behind each step, and provide helpful tips to ensure you master this technique. This thorough exploration will equip you with the knowledge and confidence to tackle similar factoring problems with ease.

Understanding Quadratic Expressions

Before diving into the specifics of factoring 5p^2 - p - 18, let's establish a solid understanding of quadratic expressions in general. A quadratic expression is a polynomial expression of the form ax^2 + bx + c, where a, b, and c are constants, and x is a variable. The term ax^2 is the quadratic term, bx is the linear term, and c is the constant term.

In our case, the expression 5p^2 - p - 18 fits this form perfectly. Here, a = 5, b = -1, and c = -18. Note that the variable is p instead of the more common x, but the underlying structure remains the same. Understanding this structure is crucial for applying the correct factoring techniques.

Factoring a quadratic expression involves rewriting it as a product of two linear expressions. In other words, we aim to find two expressions of the form (px + q) and (rx + s) such that their product is equal to the original quadratic expression. This process can seem daunting at first, but with a systematic approach and plenty of practice, it becomes a manageable and even enjoyable task.

The Factoring Process: A Step-by-Step Guide

Now, let's get down to the nitty-gritty of factoring 5p^2 - p - 18. We'll employ a method known as the "AC method," which is a versatile technique for factoring quadratic expressions of the form ax^2 + bx + c. This method is particularly useful when the coefficient of the quadratic term (a) is not equal to 1, as is the case in our expression.

Step 1: Identify a, b, and c

The first step, as we've already done, is to identify the coefficients a, b, and c in the quadratic expression. In 5p^2 - p - 18, we have:

  • a = 5
  • b = -1
  • c = -18

These values are the foundation for the subsequent steps, so it's essential to get them right. Double-check your work to ensure accuracy before moving on.

Step 2: Calculate AC

The next step is to calculate the product of a and c, which is why this method is called the AC method. In our case:

  • AC = 5 * (-18) = -90

This value, -90, is a crucial number that we will use to find the factors that will help us break down the middle term.

Step 3: Find Two Numbers That Multiply to AC and Add Up to B

This is the heart of the AC method. We need to find two numbers that, when multiplied together, equal AC (-90), and when added together, equal b (-1). This step often involves a bit of trial and error, but a systematic approach can make it much easier.

Start by listing the factor pairs of -90. Since the product is negative, one factor must be positive, and the other must be negative. Consider the following pairs:

  • 1 and -90
  • -1 and 90
  • 2 and -45
  • -2 and 45
  • 3 and -30
  • -3 and 30
  • 5 and -18
  • -5 and 18
  • 6 and -15
  • -6 and 15
  • 9 and -10
  • -9 and 10

Now, examine each pair to see which one adds up to -1 (our b value). After reviewing the pairs, you'll find that the numbers 9 and -10 satisfy both conditions:

  • 9 * (-10) = -90
  • 9 + (-10) = -1

These are the magic numbers we've been looking for!

Step 4: Rewrite the Middle Term

With our two numbers (9 and -10) in hand, we can rewrite the middle term (-p) of the quadratic expression. Instead of -p, we'll write 9p - 10p. This gives us:

  • 5p^2 + 9p - 10p - 18

Notice that we haven't changed the value of the expression; we've simply rewritten it in a more convenient form for factoring.

Step 5: Factor by Grouping

Now we can apply the technique of factoring by grouping. We'll group the first two terms and the last two terms together:

  • (5p^2 + 9p) + (-10p - 18)

Next, we'll factor out the greatest common factor (GCF) from each group. From the first group (5p^2 + 9p), the GCF is p. Factoring out p gives us:

  • p(5p + 9)

From the second group (-10p - 18), the GCF is -2. Factoring out -2 gives us:

  • -2(5p + 9)

Now our expression looks like this:

  • p(5p + 9) - 2(5p + 9)

Notice that we have a common factor of (5p + 9) in both terms. This is a key indicator that we're on the right track.

Step 6: Factor Out the Common Factor

Finally, we factor out the common factor (5p + 9) from the entire expression:

  • (5p + 9)(p - 2)

And there you have it! We have successfully factored the quadratic expression 5p^2 - p - 18 into the product of two linear expressions: (5p + 9)(p - 2).

Verification: Expanding to Check

To ensure our factoring is correct, we can expand the factored expression and see if it matches the original quadratic expression. This is a crucial step in the factoring process, as it provides a foolproof way to verify our work.

Expanding (5p + 9)(p - 2) using the distributive property (also known as FOIL) gives us:

  • 5p * p = 5p^2
  • 5p * (-2) = -10p
  • 9 * p = 9p
  • 9 * (-2) = -18

Combining these terms, we get:

  • 5p^2 - 10p + 9p - 18
  • 5p^2 - p - 18

This is exactly the original quadratic expression, 5p^2 - p - 18, so we can confidently say that our factoring is correct.

Tips and Tricks for Factoring Quadratic Expressions

Factoring quadratic expressions can be challenging, but with the right strategies and plenty of practice, you can become proficient. Here are some tips and tricks to help you along the way:

  1. Always look for a greatest common factor (GCF) first: Before applying any other factoring techniques, check if there's a GCF that can be factored out from all the terms in the expression. This can simplify the expression and make it easier to factor.
  2. Use the AC method systematically: The AC method is a powerful tool, but it's important to use it systematically. Follow the steps carefully, and don't skip any steps. This will help you avoid errors and stay organized.
  3. Practice, practice, practice: The more you practice factoring quadratic expressions, the better you'll become. Start with simpler expressions and gradually work your way up to more complex ones.
  4. Verify your answer by expanding: Always expand the factored expression to check if it matches the original expression. This is a foolproof way to catch any errors.
  5. Don't be afraid to try different combinations: Finding the two numbers that multiply to AC and add up to b can sometimes require trial and error. Don't be afraid to try different combinations until you find the right ones.
  6. Recognize special cases: Some quadratic expressions are special cases that can be factored using specific patterns. For example, the difference of squares (a^2 - b^2) can be factored as (a + b)(a - b), and perfect square trinomials (a^2 + 2ab + b^2 or a^2 - 2ab + b^2) can be factored as (a + b)^2 or (a - b)^2, respectively. Recognizing these patterns can save you time and effort.
  7. Use online resources and tools: There are many online resources and tools available to help you practice factoring quadratic expressions. Use these resources to supplement your learning and get extra practice.

Conclusion: Mastering the Art of Factoring

Factoring quadratic expressions is a crucial skill in algebra that opens doors to solving equations, simplifying expressions, and understanding mathematical relationships. By mastering techniques like the AC method and practicing consistently, you can confidently tackle a wide range of factoring problems.

In this comprehensive guide, we've walked you through the process of factoring the quadratic expression 5p^2 - p - 18, breaking down each step and providing valuable tips and tricks. Remember, factoring is a skill that improves with practice, so don't get discouraged if you encounter challenges along the way. Embrace the process, learn from your mistakes, and celebrate your successes. With dedication and perseverance, you'll become a factoring pro in no time.

So, keep practicing, keep exploring, and keep unlocking the power of algebra! Factoring quadratic expressions is just one step on your mathematical journey, and the skills you develop along the way will serve you well in many areas of mathematics and beyond.