Solving The Quadratic Equation: X² - 10x = 24

Hey guys! Today, we're going to dive into solving a classic quadratic equation. You know, those equations with the $x^2$ term that sometimes look a little intimidating? But don't worry, we'll break it down step by step, so it's super easy to follow. Our mission: to find the solution to the equation $x^2 - 10x = 24$. Let's get started!

Understanding Quadratic Equations

First things first, let's chat about what a quadratic equation actually is. At its heart, a quadratic equation is a polynomial equation of the second degree. That fancy term “second degree” just means the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation looks like this: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a isn't zero (otherwise, it wouldn't be quadratic anymore!).

Our equation, $x^2 - 10x = 24$, might not look exactly like the general form just yet, but we'll get it there. The key thing to remember is that these equations can have up to two solutions, also known as roots or zeros. These solutions are the values of x that make the equation true. So, our goal is to find those values!

There are several ways to solve quadratic equations, and we'll use a method called factoring in this case. Factoring involves rewriting the quadratic expression as a product of two binomials (expressions with two terms). This method is super handy when it works because it turns a tricky problem into something much more manageable. Other methods include using the quadratic formula or completing the square, which are great tools to have in your mathematical toolkit.

Before we jump into solving our specific equation, it's important to understand why we care about quadratic equations in the first place. They pop up all over the place in real-world applications, from physics (like projectile motion) to engineering (designing curves and shapes) to even finance (modeling growth and decay). So, mastering these equations is not just an academic exercise; it's a practical skill that can open doors in many fields. Plus, it’s a fundamental concept in mathematics, building the groundwork for more advanced topics.

Step 1: Setting the Equation to Zero

The first crucial step in solving our equation, $x^2 - 10x = 24$, using factoring (or the quadratic formula, for that matter) is to set the equation equal to zero. Why? Because we want to find the values of x that make the entire expression equal to zero. This makes the factoring process much easier. To do this, we need to move that 24 from the right side of the equation to the left side. How do we do that? Simple! We subtract 24 from both sides of the equation.

So, let’s do it:

$x^2 - 10x = 24$

Subtract 24 from both sides:

$x^2 - 10x - 24 = 24 - 24$

This simplifies to:

$x^2 - 10x - 24 = 0$

Now, our equation is in the standard quadratic form, $ax^2 + bx + c = 0$, where a = 1, b = -10, and c = -24. Getting the equation into this form is a game-changer because it sets us up perfectly for the next step: factoring. This is a common technique in algebra, and once you get the hang of it, you'll be solving quadratic equations like a pro!

Think of this step as preparing the canvas before you start painting. By setting the equation to zero, we’ve created the right environment for the factoring magic to happen. It might seem like a small step, but it’s incredibly important. If you skip this step or try to factor the equation in its original form, you’ll likely end up with the wrong answer. So, always remember: setting the equation to zero is the golden rule for solving quadratic equations!

Step 2: Factoring the Quadratic Expression

Alright, with our equation now in the form $x^2 - 10x - 24 = 0$, it’s time to roll up our sleeves and get to the heart of the problem: factoring. Factoring might sound like a scary word, but it's really just the process of breaking down a complex expression into simpler parts. In this case, we want to rewrite the quadratic expression $x^2 - 10x - 24$ as a product of two binomials. Think of it like finding the two pieces that, when multiplied together, give us our original expression.

The key to factoring a quadratic like this is to find two numbers that meet two specific criteria:

1. They multiply to give us the constant term, which is -24 in our case.
2. They add up to the coefficient of the x term, which is -10 in our case.

So, we're on a number hunt! We need two numbers that multiply to -24 and add to -10. Let's think about the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Now, we need to consider the negative sign. Since we want the numbers to multiply to a negative number, one of them has to be negative. And since they need to add up to a negative number, the larger one should be negative. Let's try a few combinations:

• -1 and 24? No, that adds up to 23.
• -2 and 12? Nope, adds up to 10.
• -3 and 8? Close, but adds up to 5.
• -12 and 2? Bingo! These multiply to -24 and add up to -10.

So, we've found our magic numbers: -12 and 2. Now we can rewrite our quadratic expression in factored form. We use these numbers to create two binomials:

$(x - 12)(x + 2) = 0$

This is the factored form of our original quadratic expression. It might seem like we've just made things more complicated, but trust me, this is a huge step forward. We've transformed a single equation into a product of two simpler expressions that equal zero. And that sets us up perfectly for the final step: finding the solutions.

Step 3: Finding the Solutions

We've reached the final stretch! We’ve successfully factored our quadratic equation into the form $(x - 12)(x + 2) = 0$. Now comes the really cool part: using this factored form to actually find the solutions for x. This is where the Zero Product Property comes into play. This property is a fundamental concept in algebra, and it’s super helpful for solving equations like this. It basically says: if the product of two factors is zero, then at least one of those factors must be zero.

In our case, we have two factors: $(x - 12)$ and $(x + 2)$. Their product is zero. So, according to the Zero Product Property, either $(x - 12)$ must be zero, or $(x + 2)$ must be zero, or both! This gives us two separate, simpler equations to solve:

1. $x - 12 = 0$
2. $x + 2 = 0$

Let’s solve the first equation. To isolate x, we simply add 12 to both sides:

$x - 12 + 12 = 0 + 12$

This gives us:

$x = 12$

So, one solution is x = 12. Great! Now let's tackle the second equation. To isolate x, we subtract 2 from both sides:

$x + 2 - 2 = 0 - 2$

This gives us:

$x = -2$

And there we have it! Our second solution is x = -2. We've successfully found both solutions to the quadratic equation $x^2 - 10x = 24$. They are x = 12 and x = -2.

It’s always a good idea to double-check our work, especially in math. We can do this by plugging each solution back into the original equation and seeing if it holds true. Let's try x = 12:

$(12)^2 - 10(12) = 144 - 120 = 24$

Yep, it works! Now let's try x = -2:

$(-2)^2 - 10(-2) = 4 + 20 = 24$

It works too! This confirms that our solutions are correct. We've navigated through the factoring process, applied the Zero Product Property, and found the values of x that make our original equation true.

Conclusion

Woo-hoo! We did it! We successfully solved the quadratic equation $x^2 - 10x = 24$. We walked through the process step-by-step: setting the equation to zero, factoring the quadratic expression, and then using the Zero Product Property to find the solutions. We discovered that the solutions are x = 12 and x = -2.

Solving quadratic equations might seem like a daunting task at first, but as you can see, breaking it down into smaller, manageable steps makes it totally doable. Factoring is a powerful tool, and the Zero Product Property is a key concept to remember. With practice, you'll become a pro at solving these types of equations.

Quadratic equations are everywhere in math and real-world applications, so understanding how to solve them is a valuable skill. Whether you're working on a physics problem, designing a building, or even just trying to figure out how to maximize your garden space, quadratic equations can come in handy. So keep practicing, keep exploring, and keep building your math skills! You've got this!