Solving X²-3x-10=0: Factorization Method Explained

Hey guys! Today, we're diving into the world of quadratic equations and tackling a common problem: finding the roots of a quadratic equation using the factorization method. Specifically, we'll be working through the equation x² - 3x - 10 = 0. Don't worry if that looks intimidating – we'll break it down step by step so it's super easy to understand. So, let's jump right in and get those roots sorted!

Understanding Quadratic Equations and Roots

Before we jump into the solution, let's make sure we're all on the same page. A quadratic equation is a polynomial equation of the second degree. That basically means it has a term with x squared (x²) as the highest power of x. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants (numbers), and 'a' cannot be zero. In our case, the equation x² - 3x - 10 = 0 fits this form perfectly, where a = 1, b = -3, and c = -10.

Now, what are we trying to find? We're looking for the roots of the equation. The roots, also sometimes called solutions or zeros, are the values of x that make the equation true. In other words, they are the values of x that, when plugged into the equation, will make the left side equal to zero. Quadratic equations can have up to two real roots, which means there are at most two values of x that will satisfy the equation. Finding these roots is the core of what we're doing today.

There are several ways to solve quadratic equations, but one of the most common and straightforward methods is factorization. Factorization involves breaking down the quadratic expression into a product of two linear expressions (expressions with x to the power of 1). This method is particularly effective when the quadratic expression can be factored easily, as is the case with our example equation. So, buckle up, because we're about to factor our way to the roots!

The Factorization Method: A Detailed Walkthrough

Alright, let's get down to business and solve the quadratic equation x² - 3x - 10 = 0 using the factorization method. This method hinges on rewriting the quadratic expression as a product of two binomials (expressions with two terms). Here's how we do it:

Step 1: Identify the Coefficients

First things first, we need to identify the coefficients in our equation. As we mentioned earlier, our equation is in the form ax² + bx + c = 0. So, in x² - 3x - 10 = 0:

• a = 1 (the coefficient of x²)
• b = -3 (the coefficient of x)
• c = -10 (the constant term)

Identifying these coefficients is crucial because they'll guide us in the next steps.

Step 2: Find Two Numbers

This is the heart of the factorization method. We need to find two numbers that:

1. Multiply to give 'ac' (the product of a and c)
2. Add up to 'b'

In our case:

• ac = 1 * (-10) = -10
• b = -3

So, we're looking for two numbers that multiply to -10 and add up to -3. Let's think about the factors of -10. We have:

• -1 and 10
• 1 and -10
• -2 and 5
• 2 and -5

Which pair adds up to -3? Bingo! It's 2 and -5. These are our magic numbers!

Step 3: Split the Middle Term

Now, we'll use these numbers to split the middle term (-3x) in our equation. We rewrite -3x as the sum of 2x and -5x. So, our equation becomes:

x² + 2x - 5x - 10 = 0

Notice that we haven't changed the equation's value; we've just rewritten it in a more useful form for factorization.

Step 4: Factor by Grouping

This is where the grouping comes in. We'll group the first two terms and the last two terms together:

(x² + 2x) + (-5x - 10) = 0

Now, we factor out the greatest common factor (GCF) from each group:

• From (x² + 2x), the GCF is x. Factoring it out, we get x(x + 2).
• From (-5x - 10), the GCF is -5. Factoring it out, we get -5(x + 2).

So, our equation now looks like this:

x(x + 2) - 5(x + 2) = 0

Notice something cool? We have a common factor of (x + 2) in both terms. This is a key sign that we're on the right track!

Step 5: Factor Out the Common Binomial

Since (x + 2) is a common factor, we can factor it out:

(x + 2)(x - 5) = 0

We've successfully factored our quadratic expression! We've transformed x² - 3x - 10 into the product of two binomials: (x + 2) and (x - 5).

Step 6: Solve for x

We're in the home stretch! Now, we use 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. In other words, if A * B = 0, then either A = 0 or B = 0 (or both).

Applying this to our factored equation (x + 2)(x - 5) = 0, we have two possibilities:

1. x + 2 = 0
2. x - 5 = 0

Let's solve each of these simple linear equations:

1. For x + 2 = 0, subtract 2 from both sides: x = -2
2. For x - 5 = 0, add 5 to both sides: x = 5

And there you have it! We've found the roots of our quadratic equation.

The Roots: x = -2 and x = 5

So, the roots of the quadratic equation x² - 3x - 10 = 0 are x = -2 and x = 5. This means that if we plug either -2 or 5 back into the original equation, it will be true.

Let's quickly verify this. For x = -2:

(-2)² - 3(-2) - 10 = 4 + 6 - 10 = 0

And for x = 5:

(5)² - 3(5) - 10 = 25 - 15 - 10 = 0

It works! Both values satisfy the equation, confirming that they are indeed the roots.

Tips and Tricks for Factorization

Factorization can be a bit tricky at first, but with practice, you'll get the hang of it. Here are a few tips and tricks to help you along the way:

• Practice, practice, practice: The more you factor quadratic equations, the better you'll become at recognizing patterns and quickly finding the right numbers.
• Look for common factors: Before attempting to factor a quadratic expression, always check if there's a common factor that can be factored out. This can simplify the expression and make it easier to factor further.
• Use the ac method: The method we used today, where we find two numbers that multiply to 'ac' and add up to 'b', is a reliable way to factor quadratic expressions.
• Don't be afraid to try different combinations: Sometimes, you might need to try a few different pairs of numbers before you find the ones that work. Don't get discouraged if your first attempt doesn't pan out. Keep trying!
• Check your work: After you've factored a quadratic expression, you can always check your work by multiplying the factors back together. If you get the original expression, you know you've factored correctly.

When Factorization Isn't Enough

While factorization is a powerful method, it's not always the easiest or most efficient way to solve quadratic equations. Some quadratic equations are difficult or impossible to factor using simple techniques. In these cases, we can turn to other methods, such as:

• The quadratic formula: The quadratic formula is a universal solution for finding the roots of any quadratic equation, regardless of whether it can be factored easily. It's a bit more involved than factorization, but it always works.
• Completing the square: Completing the square is another method that can be used to solve any quadratic equation. It involves rewriting the equation in a form that allows you to easily take the square root of both sides.

We might explore these methods in future discussions, but for now, let's focus on mastering the factorization method.

Conclusion: You've Got the Roots!

So, there you have it! We've successfully found the roots of the quadratic equation x² - 3x - 10 = 0 using the factorization method. We broke down the process into manageable steps, from identifying the coefficients to factoring the expression and solving for x. Remember, guys, the key to mastering factorization is practice. So, keep tackling those quadratic equations, and you'll become a pro in no time!

I hope this guide has been helpful. If you have any questions or want to explore more examples, feel free to ask. Happy solving!