Factoring: $2x^2 - 18x + 40$ - A Step-by-Step Guide
Hey guys! Let's dive into factoring the quadratic expression $2x^2 - 18x + 40$. Factoring is like reverse multiplication – we're trying to find the expressions that, when multiplied together, give us the original expression. This is a fundamental skill in algebra, and it's super useful for solving equations and simplifying expressions. Sometimes it can seem tricky, but don't worry, we'll break it down step by step. We’ll cover everything from identifying common factors to using the quadratic formula, so you’ll be a factoring pro in no time! Whether you’re tackling homework or just brushing up on your math skills, this guide will help you understand the ins and outs of factoring.
1. Identifying the Greatest Common Factor (GCF)
When you first look at an expression like $2x^2 - 18x + 40$, the initial step is to always check for a greatest common factor (GCF). This is the largest number and/or variable that divides evenly into all terms. In our expression, the coefficients are 2, -18, and 40. What's the biggest number that divides into all of these? It's 2! So, 2 is our GCF.
Let's factor out this 2 from the expression:
Factoring out the GCF simplifies the expression inside the parentheses, making it easier to work with. This is a crucial step because it reduces the complexity of the factoring process. It’s like taking out the big pieces of the puzzle first, so the rest fits together more smoothly. By identifying and factoring out the GCF, we’ve already made significant progress in our goal to factor the original expression. Always remember to check for the GCF as your first move – it can save you a lot of headaches down the road!
2. Factoring the Quadratic Trinomial
Now, let's focus on the trinomial inside the parentheses: $x^2 - 9x + 20$. This is a quadratic trinomial, which means it's in the form of $ax^2 + bx + c$, where in this case, a = 1, b = -9, and c = 20. To factor this, we need to find two numbers that:
- Multiply to give c (20)
- Add up to give b (-9)
Think of it like a puzzle: we need to find two pieces that fit both conditions. Let's list the factor pairs of 20:
- 1 and 20
- 2 and 10
- 4 and 5
Now, consider the signs. Since we need the numbers to add up to -9, we know both numbers must be negative. So, let's adjust our factor pairs:
- -1 and -20
- -2 and -10
- -4 and -5
Which pair adds up to -9? You got it! -4 and -5. These are our magic numbers. We can now rewrite the trinomial in factored form:
So, factoring a quadratic trinomial is like detective work – you’re searching for the right combination of numbers that satisfy the multiplication and addition conditions. Once you find those numbers, the trinomial practically factors itself! Keep practicing, and you’ll become a pro at spotting these number pairs. This method is super common in algebra, so mastering it will definitely pay off in the long run.
3. Putting It All Together
Okay, we've done the individual pieces, now let's assemble the final factored form. Remember we factored out the GCF, 2, in the first step? We need to include that in our final answer. So, we take the GCF and multiply it by the factored trinomial:
And there you have it! The completely factored form of $2x^2 - 18x + 40$ is $2(x - 4)(x - 5)$. It’s like building with LEGOs: first, you separate the pieces (find the GCF and factor the trinomial), then you put them back together in the right way to get the finished product. Always make sure you include any GCF you factored out at the beginning; it’s a common mistake to forget it, but it’s a crucial part of the final answer. Double-check your work by mentally multiplying the factors back together to see if you get the original expression – this will ensure you’ve factored correctly!
4. Checking Your Answer
It’s always a good idea to check your answer to make sure you factored correctly. The easiest way to do this is to multiply the factors back together and see if you get the original expression. Let's multiply $2(x - 4)(x - 5)$:
First, multiply the binomials (x - 4) and (x - 5):
Now, multiply the result by the GCF, 2:
Ta-da! We got back our original expression. This confirms that our factoring is correct. Checking your work is like proofreading a paper – it catches any small errors and ensures your final answer is spot-on. This step is especially helpful on tests or quizzes where you want to be absolutely sure of your answer. So, always take a few moments to check your work; it’s a simple habit that can make a big difference in your results!
Conclusion
So, guys, we've successfully factored the expression $2x^2 - 18x + 40$ into $2(x - 4)(x - 5)$. Remember, factoring is a fundamental skill in algebra, and it’s all about breaking down complex expressions into simpler parts. We started by identifying and factoring out the GCF, then we factored the resulting quadratic trinomial, and finally, we checked our answer to make sure we did it right. Each step is like a piece of a puzzle, and when you put them all together, you get the full picture. Keep practicing these steps, and you'll become more confident and efficient at factoring. Whether you're solving equations, simplifying expressions, or tackling more advanced math problems, mastering factoring will give you a solid foundation. Keep up the great work, and happy factoring!