Solve: 49^x - 6 * 7^x - 7 = 0. Find X!

Hey guys! Today, we're diving into solving an interesting exponential equation. Let's break it down step-by-step so you can ace similar problems in the future. Our mission is to find the value of x in the equation:

49^x - 6 * 7^x - 7 = 0

And we have some options:

a) 1 b) 2 c) 0 d) -1

Buckle up, because we're about to solve this mystery!

Step 1: Recognize the Pattern

The first thing we need to notice is that 49 is actually 7 squared (7^2). This is super important because it allows us to rewrite our equation in a more manageable form. Recognizing these kinds of patterns is key in many math problems. So, let's rewrite 49^x as (7²⁾x. Remember that when you raise a power to another power, you multiply the exponents. Therefore, (7²⁾x becomes 7^(2x). Now our equation looks like this:

7^(2x) - 6 * 7^x - 7 = 0

Step 2: Substitution – Making it Look Familiar

Now, to make things even easier, we're going to use a little trick called substitution. Let's say that y = 7^x. This might seem a bit abstract, but it’s a common technique to simplify complex equations. By substituting, we can transform our exponential equation into a quadratic equation, which we know how to solve. Replacing every instance of 7^x with y, our equation transforms into:

y^2 - 6y - 7 = 0

See? Much friendlier, right?

Step 3: Solving the Quadratic Equation

Now we have a standard quadratic equation in the form of ay^2 + by + c = 0, where a = 1, b = -6, and c = -7. There are several ways to solve a quadratic equation, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the easiest approach. We need to find two numbers that multiply to -7 and add up to -6. Those numbers are -7 and 1. So, we can factor the quadratic equation as:

(y - 7)(y + 1) = 0

This gives us two possible solutions for y:

• y - 7 = 0 => y = 7
• y + 1 = 0 => y = -1

So, we have two potential values for y: 7 and -1.

Step 4: Back to the Original Variable

Remember that we made a substitution earlier: y = 7^x. Now we need to go back and solve for x using the values we found for y. Let's start with the first value, y = 7:

7^x = 7

Since 7 is the same as 7^1, we can easily see that:

x = 1

Now let's consider the second value, y = -1:

7^x = -1

Here's where things get interesting. Can you raise a positive number (7) to any power and get a negative number (-1)? Nope! Exponential functions with a positive base will always result in a positive value. Therefore, there is no real solution for x in this case. So, we discard y = -1.

Step 5: The Solution

So, the only valid solution we found is x = 1. That means the correct answer is:

a) 1

Justification

To recap, here’s how we solved the equation:

1. Recognized that 49^x can be written as (7²⁾x or 7^(2x).
2. Substituted y = 7^x to transform the exponential equation into a quadratic equation.
3. Solved the quadratic equation y^2 - 6y - 7 = 0 by factoring to find y = 7 and y = -1.
4. Substituted back to find the values of x. We found that 7^x = 7 gives x = 1, and 7^x = -1 has no real solution.
5. Therefore, the only solution is x = 1.

Additional Insights and Tips

• Understanding Exponential Functions: Exponential functions of the form a^x, where a > 0, are always positive. This is why we discarded the solution 7^x = -1. Understanding the properties of exponential functions is crucial for solving these types of problems.

• Substitution: The substitution technique is incredibly useful for simplifying complex equations. It allows you to transform the equation into a more familiar form, such as a quadratic equation. Practice recognizing when and how to use substitution.

• Factoring Quadratics: Being proficient in factoring quadratic equations is essential. If factoring isn't your strong suit, consider practicing different factoring techniques.

• Checking Solutions: Always check your solutions by plugging them back into the original equation to ensure they are valid. In this case, plugging x = 1 into the original equation gives:

  49^1 - 6 * 7^1 - 7 = 49 - 42 - 7 = 0
  

  This confirms that x = 1 is indeed a solution.

Common Mistakes to Avoid

• Forgetting the Substitution: A common mistake is forgetting to substitute back to the original variable x after solving for y. Always remember to go back to the original variable to find the final solution.
• Ignoring Invalid Solutions: Be careful to discard any solutions that don't make sense in the context of the original equation. For example, in this case, 7^x = -1 has no real solution, so it should be discarded.
• Incorrect Factoring: Make sure to factor the quadratic equation correctly. Incorrect factoring can lead to incorrect solutions.

Conclusion

So there you have it! By recognizing patterns, using substitution, and solving the resulting quadratic equation, we found that the value of x in the equation 49^x - 6 * 7^x - 7 = 0 is 1. Keep practicing these techniques, and you'll be solving exponential equations like a pro in no time! Keep your mind sharp, and always double-check your work!