Solve The Equation:$\[ 3^{2x} - 2(3^{x+1}) + 9 = 0 \\]

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve, but with the right techniques and strategies, they can be tackled with ease. In this article, we will focus on solving the equation $3^{2x} - 2(3^{x+1}) + 9 = 0$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $3^{2x} - 2(3^{x+1}) + 9 = 0$. This equation involves an exponential function with base 3. The first step in solving this equation is to simplify it by using the properties of exponents.

Simplifying the Equation

We can start by simplifying the equation using the property of exponents that states $a^{m+n} = a^m \cdot a^n$. We can rewrite the equation as follows:

$$3^{2x} - 2 \cdot 3^x \cdot 3^1 + 9 = 0$$

Now, we can simplify the equation further by combining like terms.

Combining Like Terms

We can combine the like terms in the equation as follows:

$$3^{2x} - 6 \cdot 3^x + 9 = 0$$

Introducing a New Variable

To make the equation easier to solve, we can introduce a new variable. Let's say $y = 3^x$. This allows us to rewrite the equation in terms of $y$.

Rewriting the Equation

We can rewrite the equation in terms of $y$ as follows:

$$y^2 - 6y + 9 = 0$$

Solving the Quadratic Equation

The equation $y^2 - 6y + 9 = 0$ is a quadratic equation. We can solve this equation using the quadratic formula or by factoring.

Factoring the Quadratic Equation

We can factor the quadratic equation as follows:

$$(y - 3)^2 = 0$$

Solving for $y$

We can solve for $y$ by setting the expression inside the parentheses equal to zero.

$$y - 3 = 0$$

Solving for $x$

Now that we have found the value of $y$, we can substitute it back into the equation $y = 3^x$ to solve for $x$.

$$3^x = 3$$

Solving for $x$

We can solve for $x$ by taking the logarithm of both sides of the equation.

$$x = \log_3(3)$$

Simplifying the Expression

We can simplify the expression $\log_3(3)$ by using the property of logarithms that states $\log_a(a) = 1$.

$$x = 1$$

Conclusion

In this article, we have solved the equation $3^{2x} - 2(3^{x+1}) + 9 = 0$ using a step-by-step approach. We have introduced a new variable, rewritten the equation in terms of the new variable, and solved the resulting quadratic equation. We have also solved for $x$ by taking the logarithm of both sides of the equation. The final answer is $x = 1$.

Real-World Applications

Exponential equations have many real-world applications. For example, they can be used to model population growth, chemical reactions, and financial investments. In the field of finance, exponential equations can be used to calculate compound interest and investment returns.

Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

• Use the properties of exponents: Exponential equations involve exponents, so it's crucial to use the properties of exponents to simplify the equation.
• Introduce a new variable: Introducing a new variable can make the equation easier to solve.
• Use the quadratic formula: The quadratic formula can be used to solve quadratic equations.
• Take the logarithm: Taking the logarithm of both sides of the equation can help solve for the variable.

Common Mistakes

When solving exponential equations, it's essential to avoid the following common mistakes:

• Not using the properties of exponents: Failing to use the properties of exponents can make the equation difficult to solve.
• Not introducing a new variable: Failing to introduce a new variable can make the equation difficult to solve.
• Not using the quadratic formula: Failing to use the quadratic formula can make it difficult to solve the quadratic equation.
• Not taking the logarithm: Failing to take the logarithm of both sides of the equation can make it difficult to solve for the variable.

Conclusion

Introduction

In our previous article, we discussed how to solve exponential equations using a step-by-step approach. We introduced a new variable, rewrote the equation in terms of the new variable, and solved the resulting quadratic equation. We also solved for $x$ by taking the logarithm of both sides of the equation. In this article, we will answer some frequently asked questions about solving exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. Exponential functions have the form $a^x$, where $a$ is the base and $x$ is the exponent.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents. For example, you can rewrite the equation $3^{2x} - 2(3^{x+1}) + 9 = 0$ as $3^{2x} - 6 \cdot 3^x + 9 = 0$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, you can plug these values into the quadratic formula to solve for $x$.

Q: What is the logarithm?

A: The logarithm is the inverse of the exponential function. It is a mathematical operation that takes a number and returns the exponent to which a base must be raised to produce that number.

Q: How do I use the logarithm to solve an exponential equation?

A: To use the logarithm to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to solve for the variable.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not using the properties of exponents
• Not introducing a new variable
• Not using the quadratic formula
• Not taking the logarithm

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Real-World Applications

Exponential equations have many real-world applications. For example, they can be used to model population growth, chemical reactions, and financial investments. In the field of finance, exponential equations can be used to calculate compound interest and investment returns.

Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

• Use the properties of exponents: Exponential equations involve exponents, so it's crucial to use the properties of exponents to simplify the equation.
• Introduce a new variable: Introducing a new variable can make the equation easier to solve.
• Use the quadratic formula: The quadratic formula can be used to solve quadratic equations.
• Take the logarithm: Taking the logarithm of both sides of the equation can help solve for the variable.

Conclusion

In conclusion, solving exponential equations requires a step-by-step approach. By using the properties of exponents, introducing a new variable, and solving the resulting quadratic equation, we can solve exponential equations with ease. Remember to use the quadratic formula and take the logarithm of both sides of the equation to solve for the variable. With practice and patience, you can become proficient in solving exponential equations.

Additional Resources

For more information on solving exponential equations, you can check out the following resources:

• Online tutorials: Websites such as Khan Academy and Mathway offer online tutorials and practice exercises on solving exponential equations.
• Textbooks: Textbooks such as "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by James Stewart provide detailed explanations and examples of solving exponential equations.
• Practice tests: Practice tests such as the ACT and SAT math sections include exponential equations as part of the test.

Final Thoughts

Solving exponential equations can be challenging, but with practice and patience, you can become proficient in solving them. Remember to use the properties of exponents, introduce a new variable, and solve the resulting quadratic equation. With the right tools and resources, you can master the art of solving exponential equations.