Factoring U^2 - 17u + 16 = 0: A Step-by-Step Guide

by SLV Team 51 views

Hey guys! Today, we're diving into the world of quadratic equations. Specifically, we're going to tackle the equation u² - 17u + 16 = 0 and break down how to factor it. Factoring quadratic equations might seem daunting at first, but with a clear, step-by-step approach, it becomes a whole lot easier. So, let's get started and make sure you understand every single step! This guide is designed to help you not only solve this specific equation but also equip you with the skills to factor similar quadratics in the future.

Understanding Quadratic Equations

Before we jump into factoring, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. In our case, the equation u² - 17u + 16 = 0 fits this form perfectly. Here, 'a' is 1, 'b' is -17, and 'c' is 16. Understanding this standard form is crucial because it provides a framework for identifying the coefficients that we'll use in the factoring process.

The Importance of Factoring

Why do we even bother factoring quadratic equations? Well, factoring is a powerful technique for solving these equations. When we factor a quadratic equation, we rewrite it as a product of two binomials. Setting each binomial equal to zero allows us to find the values of the variable that make the entire equation true. These values are also known as the roots or solutions of the equation. Factoring is not just an algebraic exercise; it's a key method for solving real-world problems in physics, engineering, and economics. Mastering factoring techniques opens the door to understanding more complex mathematical concepts and their applications.

Step-by-Step Factoring Process

Now, let's get down to the nitty-gritty of factoring our equation u² - 17u + 16 = 0. We'll break it down into manageable steps to make sure you've got it.

Step 1: Identify the Coefficients

The first thing we need to do is identify the coefficients 'a', 'b', and 'c' in our equation. As we mentioned earlier, in u² - 17u + 16 = 0, we have:

  • a = 1
  • b = -17
  • c = 16

Identifying these coefficients correctly is the foundation for the rest of the factoring process. Make sure you pay close attention to the signs (positive or negative) because they play a significant role in determining the factors.

Step 2: Find Two Numbers

This is where the real factoring magic happens. We need to find two numbers that:

  1. Multiply to 'c' (which is 16 in our case).
  2. Add up to 'b' (which is -17 in our case).

This might sound like a puzzle, and it kind of is! To find these numbers, we can start by listing the factor pairs of 16. These are the pairs of numbers that multiply together to give 16:

  • 1 and 16
  • 2 and 8
  • 4 and 4
  • -1 and -16
  • -2 and -8
  • -4 and -4

Now, we need to check which of these pairs adds up to -17. Looking at the list, we can see that -1 and -16 fit the bill. They multiply to 16 (-1 * -16 = 16) and add up to -17 (-1 + -16 = -17). These are our magic numbers!

Step 3: Rewrite the Middle Term

Now that we've found our two numbers, -1 and -16, we can rewrite the middle term (-17u) in our equation using these numbers. This is a crucial step in the factoring process. Instead of -17u, we'll write -1u - 16u. Our equation now looks like this:

u² - 1u - 16u + 16 = 0

Notice that we haven't changed the value of the equation; we've simply rewritten it in a more useful form for factoring. This technique allows us to break the quadratic expression into smaller, more manageable parts.

Step 4: Factor by Grouping

Here comes the fun part – factoring by grouping! We'll group the first two terms and the last two terms together:

(u² - 1u) + (-16u + 16) = 0

Now, we'll factor out the greatest common factor (GCF) from each group. In the first group (u² - 1u), the GCF is 'u'. Factoring out 'u', we get:

u(u - 1)

In the second group (-16u + 16), the GCF is -16. Factoring out -16, we get:

-16(u - 1)

So, our equation now looks like this:

u(u - 1) - 16(u - 1) = 0

Notice something cool? Both terms now have a common factor of (u - 1). This is a sign that we're on the right track!

Step 5: Factor Out the Common Binomial

Since both terms in our equation have the common binomial factor (u - 1), we can factor it out. This is the final step in the factoring process. Factoring out (u - 1), we get:

(u - 1)(u - 16) = 0

And there you have it! We've successfully factored the quadratic equation u² - 17u + 16 = 0 into the product of two binomials: (u - 1)(u - 16). This is the factored form of our equation, and it's the key to finding the solutions.

Finding the Solutions

Now that we've factored the equation, finding the solutions is a piece of cake. To find the solutions, we set each factor equal to zero and solve for 'u'. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

Step 6: Set Each Factor to Zero

We have two factors: (u - 1) and (u - 16). Setting each to zero, we get:

  1. u - 1 = 0
  2. u - 16 = 0

Step 7: Solve for 'u'

Now, we simply solve each equation for 'u'.

For the first equation, u - 1 = 0, we add 1 to both sides:

u = 1

For the second equation, u - 16 = 0, we add 16 to both sides:

u = 16

So, the solutions to the equation u² - 17u + 16 = 0 are u = 1 and u = 16. These are the values of 'u' that make the equation true. We've not only factored the equation but also found its roots!

Let's Summarize

Okay, guys, let's quickly recap the steps we took to factor the quadratic equation u² - 17u + 16 = 0:

  1. Identify the Coefficients: a = 1, b = -17, c = 16.
  2. Find Two Numbers: Two numbers that multiply to 16 and add up to -17 (-1 and -16).
  3. Rewrite the Middle Term: u² - 1u - 16u + 16 = 0.
  4. Factor by Grouping: u(u - 1) - 16(u - 1) = 0.
  5. Factor Out the Common Binomial: (u - 1)(u - 16) = 0.
  6. Set Each Factor to Zero: u - 1 = 0 and u - 16 = 0.
  7. Solve for 'u': u = 1 and u = 16.

By following these steps, you can factor a wide range of quadratic equations. Remember, practice makes perfect, so don't be afraid to tackle more examples!

Additional Tips for Factoring

Here are a few extra tips to help you become a factoring pro:

  • Always look for a greatest common factor (GCF) first: If the terms in the equation have a common factor, factoring it out can simplify the equation and make it easier to factor.
  • Practice identifying number pairs: The more you practice finding pairs of numbers that multiply to 'c' and add up to 'b', the quicker you'll become at factoring.
  • Don't give up! Factoring can be challenging, but with persistence, you'll get the hang of it. Try different approaches and don't be afraid to make mistakes. Mistakes are learning opportunities!

Conclusion

Factoring the quadratic equation u² - 17u + 16 = 0 might have seemed like a daunting task at first, but by breaking it down into manageable steps, we've shown that it's totally achievable. We've not only factored the equation but also found its solutions, giving you a complete understanding of the process. Remember, the key to mastering factoring is practice. So, keep practicing, and you'll become a factoring whiz in no time!

I hope this guide has been helpful for you guys. If you have any questions or want to tackle more factoring challenges, feel free to drop a comment below. Happy factoring!