Factoring Quadratics: Find The Factors Of F(x) = 8x² + 2x - 15

Hey math enthusiasts! Today, we're diving into the world of quadratic equations and, more specifically, how to find the factors of a given quadratic function. Our star of the show is the function $f(x) = 8x^2 + 2x - 15$. Don't worry, we'll break it down step by step, making sure everyone understands the process. Factoring might seem a bit intimidating at first, but with a little practice, you'll be able to solve it like a pro. So, grab your pencils and let's get started!

Understanding Quadratic Functions and Factoring

First things first, what exactly is a quadratic function? Well, it's a function that can be written in the form of $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These functions create those cool U-shaped curves called parabolas when graphed. Factoring, on the other hand, is the process of breaking down a quadratic expression into a product of simpler expressions, usually binomials (expressions with two terms). Think of it like this: if you have a number like 12, you can factor it into 3 and 4 (since 3 * 4 = 12). Factoring a quadratic is similar, but instead of numbers, we're dealing with expressions.

The main goal of factoring is to rewrite the quadratic expression in a way that makes it easier to solve for the roots (the values of x where the function equals zero). These roots are also known as the x-intercepts of the parabola. There are several methods for factoring quadratic equations, including trial and error, the AC method, and using the quadratic formula (although, for this particular problem, factoring is the most straightforward approach). Remember, the fundamental idea behind factoring is to reverse the process of multiplication (expanding). When you multiply two binomials together, you're essentially expanding them. Factoring is like going backward; you are trying to find the binomials that, when multiplied, give you the original quadratic expression. This skill is super valuable not only in math class but also in a variety of real-world applications, such as physics and engineering.

Now, let's look at our function: $f(x) = 8x^2 + 2x - 15$. We need to find two binomials that, when multiplied together, will equal this expression. The first thing we need to do is to consider the factors of the first and last terms of the quadratic function. This will help us to narrow down the possible options.

Step-by-Step Guide to Factoring $8x^2 + 2x - 15$

Alright, let's get down to business and figure out how to factor $f(x) = 8x^2 + 2x - 15$. There are several ways to approach this, but we'll stick to a method that combines logical deduction with a bit of trial and error. This method is often called the trial-and-error method, and here's how it works:

1. Look at the leading term (8x²): The factors of $8x^2$ could be $8x$ and $x$ or $4x$ and $2x$. We'll keep both options in mind, and decide which to use later. It also tells us that the product of the first terms in our binomials will need to be $8x^2$.
2. Look at the constant term (-15): The factors of -15 are pairs such as: -1 and 15, 1 and -15, -3 and 5, and 3 and -5. Since the product of the constant terms in our binomials needs to be -15, we'll use these factors to find the correct combinations.
3. Combine and Test: Now, we'll try different combinations of these factors. This is where the trial and error comes in. Our goal is to find two binomials that, when multiplied out, give us the original expression ($8x^2 + 2x - 15$).
   • Let's consider the possible factors of the quadratic function. The possible binomial factors are: $(4x + a)(2x + b)$, where a and b are the factors of -15.
   • To get the middle term (+2x), we'll have to play with the factors of -15. Remember, when we multiply the binomials out, the outer and inner terms will combine to give us our middle term.

Let's test the answer options:

• Option A: (4x + 5)(2x - 3)
  • Expanding this gives us $8x^2 - 12x + 10x - 15 = 8x^2 - 2x - 15$. This is not correct because the middle term is -2x, not +2x.
• Option B: (4x + 3)(2x + 3)
  • Expanding this gives us $8x^2 + 12x + 6x + 9 = 8x^2 + 18x + 9$. This is not correct.
• Option C: (4x - 3)(2x + 5)
  • Expanding this gives us $8x^2 + 20x - 6x - 15 = 8x^2 + 14x - 15$. This is not correct.
• Option D: (4x - 5)(2x + 3)
  • Expanding this gives us $8x^2 + 12x - 10x - 15 = 8x^2 + 2x - 15$. This is correct.

So, by carefully selecting the correct factors and then expanding the binomials to make sure they match our original expression, we have found our answer.

The Correct Answer and Why

After going through each option, we've found that the correct factorization of $8x^2 + 2x - 15$ is (4x - 5)(2x + 3). When we expand this, we get:

$(4x - 5)(2x + 3) = 8x^2 + 12x - 10x - 15 = 8x^2 + 2x - 15$

This matches our original function perfectly! The other options did not result in the original quadratic equation. Therefore, the factors of the function $f(x) = 8x^2 + 2x - 15$ are (4x - 5) and (2x + 3). Remember, when factoring, always double-check your work by expanding the factored form to make sure you end up with the original expression. This extra step can save you a lot of time and effort.

Tips and Tricks for Factoring

Factoring can be tricky, but here are some tips to make the process smoother:

• Practice, practice, practice! The more you factor, the easier it becomes. Try different examples and work through the steps.
• Always look for a greatest common factor (GCF) first. Sometimes, you can simplify the expression immediately by factoring out a GCF from all terms.
• Master the signs. Pay close attention to the signs (+ or -) in the original expression and how they affect the factors.
• Use the AC method or other methods If trial and error feels overwhelming, try other systematic methods like the AC method. There are many ways to skin the cat!
• Check your work! Always expand your factored form to ensure it matches the original expression.

Final Thoughts

Congratulations, guys! You've successfully factored a quadratic equation. Keep practicing, and you'll become more confident in this skill. Factoring is an important concept in algebra, so it's worth the effort to master it. Now go forth and conquer those quadratic equations!