Solving X In The Equation X² + 10x + 12 = 36

by ADMIN 45 views

Hey guys! Let's dive into a classic math problem today: solving for x in the equation x² + 10x + 12 = 36. This type of quadratic equation pops up everywhere, from basic algebra to more advanced calculus, so mastering it is super important. We'll break it down step-by-step, making sure you understand not just how to solve it, but why each step works. So, grab your pencils, and let's get started!

1. Setting Up the Quadratic Equation

Before we jump into any formulas, the first thing we need to do is get our equation into the standard quadratic form. Remember, the standard form looks like this: ax² + bx + c = 0. Right now, our equation is x² + 10x + 12 = 36. See how it's not quite in the standard form because of that 36 on the right side? Our mission is to make that side zero.

Why is this important, you ask? Well, having the equation in standard form allows us to easily identify the coefficients a, b, and c, which are crucial for using methods like the quadratic formula or factoring. Think of it as preparing our ingredients before we start cooking – you wouldn't try to bake a cake without measuring out your flour, right? Similarly, we need this standard form to solve the equation effectively.

So, how do we get there? It's simple: we subtract 36 from both sides of the equation. This keeps the equation balanced (what we do to one side, we must do to the other) and moves us closer to our goal. Let's do it:

x² + 10x + 12 - 36 = 36 - 36

This simplifies to:

x² + 10x - 24 = 0

Bingo! We've successfully transformed our equation into the standard quadratic form. Now we can clearly see that a = 1, b = 10, and c = -24. These values are the keys we'll need to unlock the solutions for x.

With the equation in this neat and tidy form, we're ready to explore our options for solving it. We have two main paths we can take: factoring or using the quadratic formula. Let's delve into factoring first and see if it's a good fit for our problem.

2. Solving by Factoring

Okay, let's talk factoring. Factoring is like reverse multiplication – we're trying to find two binomials (expressions with two terms) that, when multiplied together, give us our quadratic equation. It's often the quickest way to solve a quadratic equation, but it doesn't always work. Some equations are just too stubborn to be factored easily.

So, how do we factor x² + 10x - 24 = 0? We need to find two numbers that:

  • Multiply to give us c (-24 in this case)
  • Add up to give us b (10 in this case)

This might sound like a puzzle, and that's because it is! It's a fun little mental exercise. Let's think about the factors of -24. We could have:

  • -1 and 24
  • -2 and 12
  • -3 and 8
  • -4 and 6

And their negative counterparts, of course. Now, which of these pairs adds up to 10? Ah, ha! It's -2 and 12!

So, we can rewrite our quadratic equation in factored form like this:

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

Awesome! We've factored the quadratic. But we're not done yet. Remember, we're trying to find the values of x that make this equation true. For the product of two things to be zero, at least one of them must be zero. This is a crucial concept called the zero product property. It's the key that unlocks our solutions.

So, we set each factor equal to zero and solve for x:

  • x - 2 = 0 => x = 2
  • x + 12 = 0 => x = -12

There you have it! We've found our solutions by factoring: x = 2 and x = -12. Factoring worked beautifully in this case, but what if it hadn't? That's where our next method, the quadratic formula, comes in to save the day. It's a bit more involved, but it works every time, no matter how messy the equation.

3. Solving by Using the Quadratic Formula

Alright, let's talk about the quadratic formula. This is like the Swiss Army knife of quadratic equations – it might not always be the fastest tool, but it'll get the job done every single time. Even when factoring seems impossible, the quadratic formula is your reliable backup.

So, what exactly is the quadratic formula? Buckle up, because here it comes:

x = (-b ± √(b² - 4ac)) / 2a

Woah, that looks like a mouthful, right? Don't worry, it's not as scary as it seems. It's just a formula that uses the coefficients a, b, and c from our standard quadratic equation (ax² + bx + c = 0) to calculate the solutions for x. Remember those values we identified earlier? This is where they really shine.

Let's plug in our values from the equation x² + 10x - 24 = 0. We have a = 1, b = 10, and c = -24. Substituting these into the formula, we get:

x = (-10 ± √(10² - 4 * 1 * -24)) / (2 * 1)

Now, let's simplify this step by step. First, we'll tackle what's inside the square root:

10² - 4 * 1 * -24 = 100 + 96 = 196

So, our equation now looks like this:

x = (-10 ± √196) / 2

The square root of 196 is 14, so we have:

x = (-10 ± 14) / 2

See? We're getting there! The