Factoring X^2 + 3x - 40: A Step-by-Step Guide

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Factoring the Quadratic Equation: x^2 + 3x - 40 = 0

Hey guys! Today, we're diving into the world of factoring quadratic equations, specifically the equation 0 = x^2 + 3x - 40. Factoring might seem daunting at first, but trust me, it's like solving a puzzle. Once you get the hang of it, it becomes super satisfying. We'll break down each step, making sure you understand the why behind the how. So, let's get started and turn this equation into a product of two binomials.

Understanding Quadratic Equations

Before we jump into the factoring process, let's quickly recap what a quadratic equation is. A quadratic equation is essentially a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Recognizing this form is the first step in tackling any quadratic equation. Think of it as learning the alphabet before you write a word; it's that fundamental.

In our equation, 0 = x^2 + 3x - 40, we can easily identify the coefficients: a = 1, b = 3, and c = -40. These numbers are the keys to unlocking the factored form. Understanding their roles will make the entire process smoother. For example, the 'c' term, which is -40 here, gives us a crucial clue about the numbers we're looking for in the factors. It's like having a secret code that guides you to the solution. Mastering this identification process is crucial, guys. It sets the stage for successful factoring and makes more complex algebraic manipulations much easier to handle. So, remember the standard form and practice identifying a, b, and c – it’s your foundation for quadratic equation mastery.

The Factoring Process: Finding the Right Numbers

Now, let’s get to the heart of the matter: factoring. The goal here is to rewrite the quadratic equation as a product of two binomials. In other words, we want to express x^2 + 3x - 40 in the form (x + p)(x + q), where p and q are numbers we need to find. This is where the fun begins! Think of it as a detective game where you're hunting for the right clues.

The key to finding p and q lies in their relationship with the coefficients b and c from our quadratic equation. Remember, b is the coefficient of the x term (which is 3 in our case), and c is the constant term (which is -40). Here’s the crucial rule: we need to find two numbers, p and q, that multiply to c (-40) and add up to b (3). This is like hitting the jackpot in our puzzle game. Finding these numbers is the linchpin of the factoring process.

Let’s break this down. We need factors of -40 that, when added, give us 3. This means one of the factors must be positive, and the other must be negative since their product is negative. Start by listing the factor pairs of 40: (1, 40), (2, 20), (4, 10), and (5, 8). Now, consider the signs to get a sum of 3. After a little trial and error, you'll see that 8 and -5 fit the bill perfectly. 8 multiplied by -5 equals -40, and 8 plus -5 equals 3. Bingo! We’ve found our p and q. This step-by-step approach makes the search manageable and less intimidating. Remember, practice makes perfect, guys! The more you factor, the quicker you’ll become at spotting these number pairs. So, keep practicing, and you'll be a factoring pro in no time!

Constructing the Factored Form

Alright, we've done the hard work of finding the numbers that multiply to -40 and add up to 3. We found that 8 and -5 fit the bill perfectly. Now comes the satisfying part: putting it all together to construct the factored form of our equation. Remember, our goal is to express x^2 + 3x - 40 in the form (x + p)(x + q). We’ve identified p and q as 8 and -5, respectively. So, we simply plug these values into our binomial factors.

This means our factored form is (x + 8)(x - 5). See how easy that was? We took our mystery numbers and slotted them right into place. This is where all the previous steps come together, making the solution crystal clear. It's like fitting the last piece into a jigsaw puzzle, guys – super rewarding!

But wait, we're not quite done yet. It’s always a good idea to double-check our work to make sure we haven’t made any mistakes. One of the best ways to do this is by expanding the factored form back out using the FOIL method (First, Outer, Inner, Last). This will help us verify that we indeed get back our original quadratic equation, x^2 + 3x - 40. If the expansion matches our original equation, we can confidently say we've factored it correctly. If not, no worries! It just means we need to go back and check our numbers. This verification step is crucial for accuracy and builds confidence in your factoring skills. So, let’s expand (x + 8)(x - 5) and make sure we’re on the right track.

Verification: Expanding the Factored Form

So, we've got our factored form: (x + 8)(x - 5). Now, let’s put it to the test and make sure it actually expands back to our original equation, x^2 + 3x - 40. As we talked about earlier, we'll use the FOIL method to do this. FOIL stands for First, Outer, Inner, Last, and it's a handy mnemonic for making sure we multiply each term in the first binomial by each term in the second binomial.

  • First: Multiply the first terms in each binomial: x * x = x^2
  • Outer: Multiply the outer terms: x * -5 = -5x
  • Inner: Multiply the inner terms: 8 * x = 8x
  • Last: Multiply the last terms: 8 * -5 = -40

Now, let’s add these products together: x^2 - 5x + 8x - 40. Next, we combine like terms. We have -5x and 8x, which combine to give us 3x. So, our expanded form is x^2 + 3x - 40. Guess what? It matches our original equation! This means we've successfully factored the quadratic equation. Give yourselves a pat on the back, guys! This is a huge step in mastering quadratic equations.

This verification step isn't just about checking your answer; it's about deepening your understanding of factoring. By expanding the factored form, you reinforce the connection between the factors and the original quadratic expression. It’s like seeing the whole picture instead of just the pieces. So, always make it a habit to verify your factored forms – it's a game-changer for both accuracy and understanding.

The Final Solution

After all our hard work, we've finally arrived at the final solution. We started with the quadratic equation 0 = x^2 + 3x - 40 and successfully factored it into the form 0 = (x + 8)(x - 5). This is the moment where everything clicks into place, and you can see the power of factoring.

So, to answer the original question, the factored form of the equation 0 = x^2 + 3x - 40 is indeed 0 = (x + 8)(x - 5). This means we've taken a quadratic expression and broken it down into its building blocks, which are two simpler binomial expressions. Factoring is a fundamental skill in algebra, guys, and it opens doors to solving a wide range of problems. From finding the roots of a quadratic equation to simplifying complex algebraic expressions, factoring is a tool you'll use again and again.

But we don't have to stop here! We can actually take this factored form and use it to find the solutions, or roots, of the equation. Remember, the roots are the values of x that make the equation true. In the next section, we'll see how our factored form makes finding these roots a breeze. So, let’s keep going and unlock the full potential of factoring!

Finding the Solutions (Roots) of the Equation

Okay, we've successfully factored the equation 0 = x^2 + 3x - 40 into 0 = (x + 8)(x - 5). Now, let's take it a step further and find the solutions, also known as the roots, of this equation. The roots are the values of x that make the equation true, and finding them is often the ultimate goal when dealing with quadratic equations. The factored form makes this process incredibly straightforward thanks to a principle called the Zero Product Property.

The Zero Product Property is a cornerstone of algebra, and it’s super useful. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both). This might seem obvious, but it's a powerful tool for solving factored equations.

Applying this to our equation, 0 = (x + 8)(x - 5), we can say that either (x + 8) = 0 or (x - 5) = 0. This gives us two simple linear equations to solve. Let's solve them one by one.

  • For (x + 8) = 0, we subtract 8 from both sides to get x = -8. This is our first solution.
  • For (x - 5) = 0, we add 5 to both sides to get x = 5. This is our second solution.

So, the solutions (or roots) of the equation 0 = x^2 + 3x - 40 are x = -8 and x = 5. We found these values by leveraging the factored form and the Zero Product Property. Isn’t that neat? This shows how factoring isn’t just a mathematical trick; it’s a pathway to finding the actual answers to our equation. Understanding this connection between factoring and finding solutions is key to mastering algebra, guys. So, remember the Zero Product Property – it’s your secret weapon for solving factored equations!

Conclusion: The Power of Factoring

Wow, we've come a long way! We started with a quadratic equation, 0 = x^2 + 3x - 40, and we’ve taken it through the entire factoring process. We identified the coefficients, found the right numbers, constructed the factored form 0 = (x + 8)(x - 5), verified our solution, and even found the roots of the equation: x = -8 and x = 5. That’s a complete journey through factoring, and you guys nailed it!

Factoring might have seemed tricky at first, but hopefully, you now see it as a manageable and even enjoyable puzzle. The ability to factor quadratic equations is a fundamental skill in algebra, and it opens the door to solving many different types of problems. Whether you’re simplifying expressions, solving equations, or even working with graphs of quadratic functions, factoring is a tool you'll use time and time again. It’s like having a master key that unlocks many mathematical doors.

The key takeaways from this exercise are:

  • Understanding the standard form of a quadratic equation.
  • Finding the numbers that multiply to c and add up to b.
  • Constructing the factored form using these numbers.
  • Verifying your factored form by expanding.
  • Using the Zero Product Property to find the solutions (roots) of the equation.

Remember, guys, practice makes perfect. The more you work with factoring, the more confident and proficient you'll become. So, keep tackling those quadratic equations, and you'll be a factoring whiz in no time! Great job today, and happy factoring!