Factorising 2p^2 - P - 10 A Step-by-Step Guide

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Hey guys! 👋 Today, we're diving deep into the fascinating world of factorising quadratic expressions, specifically tackling the expression 2p2−p−10{2p^2 - p - 10}. If you've ever felt a little lost when you see these types of problems, don't worry – you're in the right place! We're going to break it down step-by-step, so by the end of this guide, you'll be factorising quadratics like a pro. 💪

What are Quadratic Expressions?

Before we jump into the nitty-gritty, let's quickly recap what quadratic expressions actually are. In the simplest terms, a quadratic expression is a polynomial expression where the highest power of the variable is 2. The general form of a quadratic expression is ax2+bx+c{ax^2 + bx + c}, where a{a}, b{b}, and c{c} are constants, and x{x} is the variable. Think of it as a curve (a parabola, to be precise) when you graph it. 📈

In our case, we have 2p2−p−10{2p^2 - p - 10}. Notice how the highest power of p{p} is 2? That's our clue that it's a quadratic expression. Here, a=2{a = 2}, b=−1{b = -1}, and c=−10{c = -10}. Understanding this basic form is the first step to mastering factorisation. It's like knowing the ingredients before you bake a cake! 🎂

Why Factorise?

Okay, so we know what quadratics are, but why bother factorising them? Great question! Factorising is like reverse engineering – it's the process of breaking down a complex expression into simpler parts (its factors). Think of it as taking a mixed-up puzzle and putting it back together. 🧩

Factorising helps us in several ways:

  1. Solving Quadratic Equations: One of the main reasons we factorise is to solve quadratic equations. When a quadratic expression is equal to zero, factorising it allows us to find the values of the variable that make the equation true. These values are also known as the roots or zeros of the quadratic equation. This is super useful in many real-world applications, from physics to engineering! 🚀
  2. Simplifying Expressions: Sometimes, we need to simplify complex algebraic expressions. Factorising can help us cancel out common factors and make the expression easier to work with. It’s like decluttering your room – making things neat and tidy! 🧹
  3. Graphing Functions: Understanding the factors of a quadratic expression can help us sketch its graph. The factors tell us where the parabola intersects the x-axis, which is a crucial piece of information. Think of it as having a treasure map that leads you to the important spots! 🗺️

So, as you can see, factorising is a powerful tool in our mathematical toolkit. Now, let's get our hands dirty and see how to factorise 2p2−p−10{2p^2 - p - 10}.

The Factorisation Process: A Step-by-Step Guide

Alright, let's dive into the heart of the matter: how do we actually factorise 2p2−p−10{2p^2 - p - 10}? There are a few methods out there, but we'll focus on the most common and versatile one: the 'ac' method (also known as the 'splitting the middle term' method). Trust me, once you get the hang of this, you'll be able to tackle almost any quadratic expression. 🤓

Step 1: Identify a, b, and c

As we mentioned earlier, our expression is in the form ap2+bp+c{ap^2 + bp + c}. So, let's identify our coefficients:

  • a=2{a = 2}
  • b=−1{b = -1}
  • c=−10{c = -10}

This is like gathering your ingredients before you start cooking. 🧑‍🍳

Step 2: Calculate ac

Now, we multiply a{a} and c{c}:

ac=2×−10=−20{ac = 2 \times -10 = -20}

This number, −20{-20}, is super important. We're going to use it to find the right factors. Think of it as our magic number! ✨

Step 3: Find Two Numbers That Multiply to ac and Add Up to b

This is where the puzzle-solving comes in! We need to find two numbers that:

  • Multiply to −20{-20} (our ac{ac} value)
  • Add up to −1{-1} (our b{b} value)

Let's think about the factors of −20{-20}. We could have:

  • 1 and -20
  • -1 and 20
  • 2 and -10
  • -2 and 10
  • 4 and -5
  • -4 and 5

Which pair adds up to −1{-1}? Bingo! It's 4{4} and −5{-5}. These are our golden numbers! 🌟

Step 4: Split the Middle Term

Now we take the middle term, −p{-p}, and rewrite it using our golden numbers. Instead of −p{-p}, we'll write +4p−5p{+4p - 5p}. Our expression now looks like this:

2p2+4p−5p−10{2p^2 + 4p - 5p - 10}

Notice that we haven't actually changed the value of the expression. We've just rewritten it in a clever way. It’s like rearranging the furniture in your room – it looks different, but it's still the same room! 🛋️

Step 5: Factor by Grouping

This is where the magic really happens! We're going to group the first two terms and the last two terms together:

(2p2+4p)+(−5p−10){(2p^2 + 4p) + (-5p - 10)}

Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 2p{2p}:

2p(p+2){2p(p + 2)}

From the second group, the GCF is −5{-5}:

−5(p+2){-5(p + 2)}

So, our expression now looks like this:

2p(p+2)−5(p+2){2p(p + 2) - 5(p + 2)}

Notice anything special? Both terms have a common factor of (p+2){(p + 2)}. This is our key to the final step! 🔑

Step 6: Factor Out the Common Bracket

We factor out the common bracket (p+2){(p + 2)} from the entire expression:

(p+2)(2p−5){(p + 2)(2p - 5)}

And there you have it! We've successfully factorised 2p2−p−10{2p^2 - p - 10} into (p+2)(2p−5){(p + 2)(2p - 5)}. 🎉

Checking Our Answer

It's always a good idea to check our work, right? Think of it as proofreading your essay before you submit it. 🤓 To check, we can simply expand our factors and see if we get back the original expression. Let's do it:

(p+2)(2p−5)=p(2p−5)+2(2p−5){(p + 2)(2p - 5) = p(2p - 5) + 2(2p - 5)}

=2p2−5p+4p−10{= 2p^2 - 5p + 4p - 10}

=2p2−p−10{= 2p^2 - p - 10}

Woohoo! It matches our original expression. We're golden! 🌟

Alternative Methods for Factorising

While the 'ac' method is super versatile, there are other ways to factorise quadratics. Let's briefly touch on a couple of them. Think of these as extra tools in your toolbox! 🧰

Trial and Error

This method involves a bit of educated guessing and checking. It's especially useful when the coefficients are small and the factors are easy to spot. You basically try different combinations of factors until you find one that works. It's like trying different keys on a lock until you find the right one! 🔑

For example, with 2p2−p−10{2p^2 - p - 10}, you might start by thinking about the factors of 2p2{2p^2} (which are 2p{2p} and p{p}) and the factors of −10{-10} (like 1{1} and −10{-10}, −1{-1} and 10{10}, 2{2} and −5{-5}, etc.). Then you try different combinations until you find the pair that gives you the correct middle term.

Using the Quadratic Formula

The quadratic formula is a powerful tool that can solve any quadratic equation, even if it's not easily factorisable. The formula is:

p=−b±b2−4ac2a{p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

Once you find the roots (let's call them p1{p_1} and p2{p_2}), you can write the factorised form as:

a(p−p1)(p−p2){a(p - p_1)(p - p_2)}

This method is a bit more involved, but it's a great fallback option when other methods don't work. Think of it as your secret weapon! ⚔️

Common Mistakes to Avoid

Factorising can be tricky, and it's easy to make mistakes. But don't worry, we're here to help you avoid those pitfalls! 🚧 Here are a few common mistakes to watch out for:

  1. Incorrectly Identifying a, b, and c: Make sure you have the signs right! Remember, the general form is ax2+bx+c{ax^2 + bx + c}, so b{b} and c{c} can be negative.
  2. Forgetting the Negative Sign: When finding factors that multiply to ac{ac}, remember to consider both positive and negative possibilities.
  3. Incorrectly Splitting the Middle Term: Double-check that your split terms add up to the original middle term.
  4. Not Factoring Completely: Always make sure you've factored out the greatest common factor from each group.
  5. Forgetting to Check Your Answer: As we showed earlier, checking your answer is a crucial step to ensure you haven't made any mistakes.

By being aware of these common mistakes, you'll be well on your way to factorising like a pro! 🏆

Practice Makes Perfect

Like any skill, factorising gets easier with practice. So, don't be afraid to tackle lots of different problems. The more you practice, the more confident you'll become! 🏋️‍♀️

Here are a few extra tips to help you along the way:

  • Start Simple: Begin with easier quadratics and gradually work your way up to more challenging ones.
  • Use Online Resources: There are tons of great websites and videos that can help you practice and understand factorising.
  • Ask for Help: If you're stuck, don't hesitate to ask your teacher, classmates, or an online forum for help.
  • Make It a Game: Turn factorising into a fun challenge! See how many problems you can solve in a certain amount of time.

Conclusion

So, there you have it! We've covered everything you need to know to factorise 2p2−p−10{2p^2 - p - 10} and other quadratic expressions. Remember, factorising is a valuable skill that will help you in many areas of mathematics. With practice and patience, you'll be able to tackle any quadratic that comes your way. Keep practicing, and you'll become a factorising whiz in no time! 🧙‍♂️

Thanks for joining me on this factorising adventure. Happy factoring, guys! 😊