Solve The Equation:$\[ \frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4} \\]

Introduction

In this article, we will delve into solving a complex equation involving exponents and fractions. The equation is given as: $\frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4}$. We will break down the solution into manageable steps, making it easier to understand and follow along.

Understanding the Equation

The given equation involves exponents and fractions, making it a challenging problem to solve. To start, let's simplify the equation by rewriting it in a more manageable form.

$\frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4}$

We can rewrite the equation as:

$\frac{4^x}{4} + \frac{4}{4^x} = \frac{17}{4}$

Step 1: Simplifying the Equation

To simplify the equation, we can start by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 4.

$4 \left( \frac{4^x}{4} + \frac{4}{4^x} \right) = 4 \left( \frac{17}{4} \right)$

This simplifies to:

$4^x + \frac{4}{4^x} = 17$

Step 2: Isolating the Exponential Term

Next, we can isolate the exponential term by subtracting $\frac{4}{4^x}$ from both sides of the equation.

$4^x = 17 - \frac{4}{4^x}$

Step 3: Multiplying by the Conjugate

To eliminate the fraction, we can multiply both sides of the equation by the conjugate of the right-hand side, which is $4^x$.

$4^{2x} = (17 - \frac{4}{4^x}) \cdot 4^x$

This simplifies to:

$4^{2x} = 17 \cdot 4^x - 4$

Step 4: Rearranging the Equation

We can rearrange the equation to get:

$4^{2x} - 17 \cdot 4^x + 4 = 0$

Step 5: Factoring the Quadratic

The equation is a quadratic equation in terms of $4^x$. We can factor the quadratic as:

$(4^x - 1)(4^x - 4) = 0$

Step 6: Solving for $x$

We can solve for $x$ by setting each factor equal to zero.

$4^x - 1 = 0 \Rightarrow 4^x = 1 \Rightarrow x = 0$

$4^x - 4 = 0 \Rightarrow 4^x = 4 \Rightarrow x = 1$

Conclusion

In this article, we have solved the equation $\frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4}$ using a step-by-step approach. We simplified the equation, isolated the exponential term, multiplied by the conjugate, rearranged the equation, factored the quadratic, and solved for $x$. The solutions to the equation are $x = 0$ and $x = 1$.

Final Answer

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Introduction

In our previous article, we solved the equation $\frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4}$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you better understand the solution and answer any questions you may have.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to simplify the equation, isolate the exponential term, and use algebraic manipulations to solve for $x$.

Q: Why did we multiply both sides of the equation by 4?

A: We multiplied both sides of the equation by 4 to get rid of the fractions and simplify the equation.

Q: What is the conjugate of the right-hand side of the equation?

A: The conjugate of the right-hand side of the equation is $4^x$.

Q: Why did we multiply both sides of the equation by the conjugate?

A: We multiplied both sides of the equation by the conjugate to eliminate the fraction and simplify the equation.

Q: How did we factor the quadratic equation?

A: We factored the quadratic equation by recognizing that it can be written as a product of two binomials: $(4^x - 1)(4^x - 4) = 0$.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = 0$ and $x = 1$.

Q: How can I apply this solution to real-world problems?

A: This solution can be applied to real-world problems involving exponential growth and decay. For example, if you have a population of bacteria that grows exponentially, you can use this solution to model the population growth.

Q: What are some common mistakes to avoid when solving this type of equation?

A: Some common mistakes to avoid when solving this type of equation include:

• Not simplifying the equation enough
• Not isolating the exponential term correctly
• Not using the conjugate to eliminate the fraction
• Not factoring the quadratic equation correctly

Q: How can I practice solving this type of equation?

A: You can practice solving this type of equation by working through example problems and exercises. You can also try solving similar equations with different coefficients and variables.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the solution to the equation $\frac{4^x}{4} + \frac{4}{4^x} = 4 \frac{1}{4}$. We have answered common questions and provided tips for applying the solution to real-world problems.

Final Answer

The final answer is $\boxed{0, 1}$.