Solving 2^(2x) - 12 × 2^x + 32 = 0 A Step-by-Step Guide

Hey guys! Ever stumbled upon an exponential equation that looks like it's written in a different language? Don't worry, we've all been there. Exponential equations can seem intimidating, but with the right approach, they can be broken down into manageable steps. Today, we're diving deep into solving one such equation: 2^(2x) - 12 × 2^x + 32 = 0. This isn't just about finding the answer; it's about understanding the process, so you can tackle similar problems with confidence. So, grab your thinking caps, and let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's take a moment to understand what makes an equation exponential. In simple terms, an exponential equation is an equation in which the variable appears in the exponent. Think of it like this: the power is where the action is! These equations pop up in various fields, from finance (think compound interest) to science (like radioactive decay). Recognizing them is the first step, and then comes the fun part – solving them.

Our specific equation, 2^(2x) - 12 × 2^x + 32 = 0, has a quadratic-like structure hidden within the exponential terms. This is a common trick in these types of problems. Spotting this structure is key to simplifying the equation. We need to see past the exponents and recognize the familiar pattern of a quadratic equation. This involves a clever substitution that will transform our exponential equation into something we're more comfortable with. So, keep that in mind as we move forward – spotting the hidden structure is half the battle!

Step 1: Recognizing the Quadratic Form

Okay, guys, this is where the magic happens. Our equation, 2^(2x) - 12 × 2^x + 32 = 0, might look scary, but let's rewrite it slightly. Remember that 2^(2x) is the same as (2^x)2? This little trick is crucial because it reveals the hidden quadratic form. Now our equation looks like this: (2^x)2 - 12 × 2^x + 32 = 0. See it yet? It's like a quadratic equation in disguise!

Think of it this way: if we replace 2^x with a single variable, say 'y', the equation transforms into something much simpler: y^2 - 12y + 32 = 0. This is a classic quadratic equation that we know how to solve. The key here is the substitution – it's a powerful technique for simplifying complex equations. By recognizing the quadratic form, we've turned a seemingly difficult problem into a familiar one. This is a common strategy in math: break down the complex into simpler parts, solve those parts, and then put it all back together. So, remember this trick – it'll come in handy!

Step 2: Substitution to Simplify the Equation

Alright, let's make this equation even easier to handle. As we discussed, the trick is to use substitution. We're going to let y = 2^x. This substitution is like putting on a pair of glasses – it brings the quadratic form into sharp focus. Our equation (2^x)2 - 12 × 2^x + 32 = 0 now elegantly transforms into y^2 - 12y + 32 = 0. Isn't that much friendlier?

This step is all about making the problem more accessible. By replacing the exponential term with a single variable, we've stripped away the complexity and revealed the underlying structure. It's like decluttering your room – once the mess is gone, you can see things clearly. Substitution is a fundamental technique in algebra, and it's used to simplify all sorts of equations. It allows us to work with familiar forms and apply techniques we already know. So, don't underestimate the power of a good substitution!

Step 3: Solving the Quadratic Equation

Now that we have our simplified quadratic equation, y^2 - 12y + 32 = 0, it's time to put on our quadratic-solving hats! There are a few ways to tackle this: factoring, completing the square, or using the quadratic formula. For this equation, factoring is the most straightforward approach. We need to find two numbers that multiply to 32 and add up to -12. Can you think of them?

The numbers are -4 and -8! So, we can factor the equation as (y - 4)(y - 8) = 0. This means that either y - 4 = 0 or y - 8 = 0. Solving these simple equations gives us two possible values for y: y = 4 and y = 8. We've successfully solved the quadratic equation, but remember, we're not done yet. We need to find the values of x, and for that, we need to reverse our substitution. So, let's keep those values of y in mind as we move on to the next step.

Step 4: Back-Substitution to Find x

Okay, guys, we've found the values of y, but the real question is, what are the values of x? This is where we reverse our substitution. Remember, we let y = 2^x. So, we now have two equations to solve: 2^x = 4 and 2^x = 8. These are much simpler exponential equations to deal with.

Let's take the first one: 2^x = 4. We know that 4 can be written as 2 squared (2^2), so the equation becomes 2^x = 2^2. When the bases are the same, we can equate the exponents, which gives us x = 2. Easy peasy!

Now let's tackle the second equation: 2^x = 8. Similarly, we can write 8 as 2 cubed (2^3), so the equation becomes 2^x = 2^3. Equating the exponents, we get x = 3. And there you have it! We've found two solutions for x. Back-substitution is a crucial step in solving equations where we've used substitution. It's like retracing your steps to find your way back home. Always remember to go back to the original variable to answer the question fully.

Step 5: Verifying the Solutions

Alright, we've got our solutions, x = 2 and x = 3, but before we declare victory, it's always a good idea to check our work. This is like proofreading an essay – you want to make sure everything is correct before you submit it. We'll plug our solutions back into the original equation, 2^(2x) - 12 × 2^x + 32 = 0, to see if they hold true.

Let's start with x = 2. Substituting this into the equation, we get 2^(2*2) - 12 × 2^2 + 32 = 2^4 - 12 × 4 + 32 = 16 - 48 + 32 = 0. It checks out!

Now let's try x = 3. Substituting this in, we get 2^(2*3) - 12 × 2^3 + 32 = 2^6 - 12 × 8 + 32 = 64 - 96 + 32 = 0. This one checks out too! So, we can confidently say that our solutions are correct. Verifying solutions is a key step in problem-solving. It gives you peace of mind and ensures that you haven't made any mistakes along the way. It's like the final flourish on a masterpiece!

Conclusion: Mastering Exponential Equations

So, guys, we've successfully navigated the world of exponential equations and conquered the problem 2^(2x) - 12 × 2^x + 32 = 0. We've seen how to recognize the hidden quadratic form, use substitution to simplify the equation, solve the resulting quadratic, back-substitute to find the values of x, and finally, verify our solutions. That's quite a journey!

Remember, the key to mastering exponential equations (and math in general) is practice and understanding the underlying concepts. Don't be afraid to break down complex problems into smaller, more manageable steps. Look for patterns, use substitutions, and always verify your answers. With a little bit of effort and the right approach, you can conquer any equation that comes your way. Keep practicing, keep exploring, and most importantly, keep having fun with math!

Final Answer: The solutions to the exponential equation 2^(2x) - 12 × 2^x + 32 = 0 are x = 2 and x = 3.