Solving Exponential Equations 12^(x^2 + 5x - 4) = 12^(2x + 6)

Hey there, math enthusiasts! Today, we're diving into a fascinating exponential equation that might seem intimidating at first glance, but trust me, we'll break it down step by step. We're tackling the problem: 12^(x2 + 5x - 4) = 12^(2x + 6). It looks complex, right? But don't worry, guys, we're going to solve it together in a way that's both clear and engaging. Think of this as a mathematical puzzle, and we're the detectives cracking the code! Exponential equations are a fundamental concept in algebra, and mastering them opens doors to understanding more advanced mathematical topics. So, let's get started and unlock the secrets of this equation!

Understanding Exponential Equations

Before we jump into solving our specific equation, let's take a moment to grasp the core concept of exponential equations. At their heart, exponential equations involve a variable in the exponent. The general form of an exponential equation is a^x = b, where 'a' is the base, 'x' is the exponent (our variable), and 'b' is the result. In our case, we have 12 as the base on both sides of the equation, which simplifies things a bit. Understanding this basic structure is crucial because it dictates how we approach solving these types of problems. The key to solving exponential equations lies in manipulating the equation to isolate the variable. There are several techniques we can use, including logarithms, but in this case, we'll use a more straightforward approach since the bases are the same. Remember, the power of mathematics comes from understanding the underlying principles, not just memorizing formulas. So, let's build that understanding and then apply it to our problem. Think of it like building a house – you need a strong foundation before you can put up the walls and roof.

The Golden Rule: Equal Bases, Equal Exponents

Now, here's where the magic happens! The golden rule for solving exponential equations with the same base is this: if a^m = a^n, then m = n. In simpler terms, if we have the same base raised to different powers on both sides of an equation, then those powers must be equal. This is a fundamental property of exponents and it's the key that unlocks our problem. Why is this true? Well, it stems from the fact that exponential functions are one-to-one. This means that for every unique input (exponent), there is a unique output (result). So, if two exponential expressions with the same base are equal, their exponents must be equal too. This principle allows us to transform our exponential equation into a much simpler algebraic equation, which we can then solve using familiar methods. It's like having a complex lock, but we've just found the perfect key to open it. So, let's apply this rule to our equation and see what happens. This step is crucial, so make sure you understand the logic behind it before moving on.

Applying the Rule to Our Equation

Okay, guys, let's put our golden rule into action! We have 12^(x2 + 5x - 4) = 12^(2x + 6). Notice that the base is 12 on both sides. This means we can confidently equate the exponents. So, we can rewrite the equation as x^2 + 5x - 4 = 2x + 6. See how much simpler that looks? We've transformed a potentially intimidating exponential equation into a quadratic equation, which we know how to handle. This is the beauty of mathematics – taking a complex problem and breaking it down into smaller, manageable steps. It's like climbing a mountain; you don't try to climb it all at once, you take it one step at a time. Now, let's focus on solving this quadratic equation. Our next step is to bring all the terms to one side, setting the equation equal to zero. This will allow us to use factoring or the quadratic formula to find the solutions for x. So, let's get our algebraic muscles working and rearrange those terms!

Transforming to a Quadratic Equation

Alright, let's get this quadratic equation into its standard form! We have x^2 + 5x - 4 = 2x + 6. To solve a quadratic equation, we need to set it equal to zero. So, let's subtract 2x and 6 from both sides of the equation. This gives us: x^2 + 5x - 2x - 4 - 6 = 0. Now, let's simplify by combining like terms. We have 5x - 2x = 3x and -4 - 6 = -10. So, our equation becomes: x^2 + 3x - 10 = 0. Boom! We've got a classic quadratic equation in the form ax^2 + bx + c = 0. This is a form we're familiar with, and we have several tools at our disposal to solve it. We could try factoring, completing the square, or using the quadratic formula. Factoring is often the quickest method if we can find two numbers that multiply to 'c' and add up to 'b'. So, let's see if we can factor this equation. Remember, practice makes perfect, so the more you work with quadratic equations, the easier they become. It's like learning a new language; the more you practice, the more fluent you become.

Factoring the Quadratic Equation

Now comes the fun part: factoring! We have the quadratic equation x^2 + 3x - 10 = 0. To factor this, we need to find two numbers that multiply to -10 (the constant term) and add up to 3 (the coefficient of the x term). Let's think about the factors of -10. We have pairs like 1 and -10, -1 and 10, 2 and -5, and -2 and 5. Which of these pairs adds up to 3? Bingo! It's -2 and 5. So, we can factor the quadratic equation as (x - 2)(x + 5) = 0. Isn't it satisfying when the pieces of the puzzle fall into place? Factoring is a powerful technique for solving quadratic equations, and it's often the most efficient method when it's applicable. Now that we've factored the equation, we can use the zero-product property to find the solutions for x. This property states that if the product of two factors is zero, then at least one of the factors must be zero. So, let's apply this property and find our solutions.

The Zero-Product Property: Unveiling the Solutions

Okay, guys, we're in the home stretch! We've factored our equation as (x - 2)(x + 5) = 0. Now, we use the zero-product property, which tells us that either (x - 2) = 0 or (x + 5) = 0. Let's solve each of these equations separately. For x - 2 = 0, we add 2 to both sides, giving us x = 2. For x + 5 = 0, we subtract 5 from both sides, giving us x = -5. And there you have it! We've found our solutions: x = 2 and x = -5. These are the values of x that make the original exponential equation true. It's like we've reached the summit of our mathematical mountain and can now enjoy the view. But, as good mathematicians, we should always check our solutions to make sure they work. So, let's plug these values back into the original equation and verify our answer.

Verifying the Solutions

To be absolutely sure we've nailed it, let's verify our solutions. We found x = 2 and x = -5. Let's plug these values back into the original equation: 12^(x2 + 5x - 4) = 12^(2x + 6). First, let's check x = 2. Substituting x = 2, we get: 12⁽²2 + 5(2) - 4) = 12^(2(2) + 6). Simplifying, we have 12^(4 + 10 - 4) = 12^(4 + 6), which becomes 12^10 = 12^10. This is true, so x = 2 is a valid solution. Now, let's check x = -5. Substituting x = -5, we get: 12^((-5)2 + 5(-5) - 4) = 12^(2(-5) + 6). Simplifying, we have 12^(25 - 25 - 4) = 12^(-10 + 6), which becomes 12^-4 = 12^-4. This is also true, so x = -5 is a valid solution. We've done it! We've successfully solved the exponential equation and verified our solutions. Give yourselves a pat on the back, guys! This is a testament to the power of understanding mathematical principles and applying them step by step.

Conclusion: Mastering Exponential Equations

So, there you have it! We've successfully solved the exponential equation 12^(x2 + 5x - 4) = 12^(2x + 6), and we found the solutions to be x = 2 and x = -5. We started by understanding the basics of exponential equations, then applied the golden rule of equating exponents when the bases are the same. We transformed the exponential equation into a quadratic equation, factored it, used the zero-product property, and finally, verified our solutions. This journey demonstrates the power of breaking down complex problems into smaller, manageable steps. Remember, guys, mathematics is not just about finding the right answer; it's about the process of problem-solving and the understanding you gain along the way. Keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is vast and fascinating, and there's always something new to discover. So, go forth and conquer more mathematical challenges! You've got this!