Solve The Equation: 5 X − 5 − X 2 = 2 \frac{5^x - 5^{-x}}{2} = 2 2 5 X − 5 − X ​ = 2 X ≈ X \approx X ≈ Enter Your Answer □ \square □ (Round The Answer To Three Decimal Places.)

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $\frac{5^x - 5^{-x}}{2} = 2$, which involves both positive and negative exponents. We will break down the solution into manageable steps, using algebraic manipulations and properties of exponential functions to arrive at the final answer.

Understanding the Equation

The given equation is $\frac{5^x - 5^{-x}}{2} = 2$. To solve this equation, we need to isolate the variable $x$. The first step is to get rid of the fraction by multiplying both sides of the equation by 2.

\frac{5^x - 5^{-x}}{2} = 2
\Rightarrow 5^x - 5^{-x} = 4

Using Algebraic Manipulations

Now that we have simplified the equation, we can use algebraic manipulations to isolate the variable $x$. We can start by adding $5^{-x}$ to both sides of the equation.

5^x - 5^{-x} = 4
\Rightarrow 5^x = 4 + 5^{-x}

Next, we can multiply both sides of the equation by $5^x$ to get rid of the fraction.

5^x = 4 + 5^{-x}
\Rightarrow (5^x)^2 = (4 + 5^{-x})5^x

Using Properties of Exponential Functions

Now that we have simplified the equation, we can use properties of exponential functions to isolate the variable $x$. We can start by using the property that $a^m \cdot a^n = a^{m+n}$.

(5^x)^2 = (4 + 5^{-x})5^x
\Rightarrow 5^{2x} = 4 \cdot 5^x + 5^{x-x}

Next, we can use the property that $a^m \cdot a^n = a^{m+n}$ to simplify the right-hand side of the equation.

5^{2x} = 4 \cdot 5^x + 5^{x-x}
\Rightarrow 5^{2x} = 4 \cdot 5^x + 5^0

Solving for x

Now that we have simplified the equation, we can solve for $x$. We can start by using the property that $a^m \cdot a^n = a^{m+n}$ to rewrite the equation.

5^{2x} = 4 \cdot 5^x + 5^0
\Rightarrow 5^{2x} = 4 \cdot 5^x + 1

Next, we can use the property that $a^m \cdot a^n = a^{m+n}$ to rewrite the equation.

5^{2x} = 4 \cdot 5^x + 1
\Rightarrow 5^{2x} - 4 \cdot 5^x = 1

Now, we can factor the left-hand side of the equation.

5^{2x} - 4 \cdot 5^x = 1
\Rightarrow (5^x)^2 - 4 \cdot 5^x = 1

Using the Quadratic Formula

Now that we have factored the left-hand side of the equation, we can use the quadratic formula to solve for $x$. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, we have:

$$a = 1, b = -4, c = -1$$

Plugging these values into the quadratic formula, we get:

x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)}
\Rightarrow x = \frac{4 \pm \sqrt{16 + 4}}{2}
\Rightarrow x = \frac{4 \pm \sqrt{20}}{2}
\Rightarrow x = \frac{4 \pm 2\sqrt{5}}{2}
\Rightarrow x = 2 \pm \sqrt{5}

Rounding the Answer

Finally, we need to round the answer to three decimal places. Using a calculator, we get:

$$x \approx 2.236$$

Therefore, the final answer is:

$x \approx \boxed{2.236}$

Conclusion

$$$$$$

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$ using algebraic manipulations and properties of exponential functions. This may involve multiplying both sides of the equation by a power of the base, adding or subtracting the same value to both sides, or using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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

A: To use the quadratic formula to solve an exponential equation, you need to rewrite the equation in the form $ax^2 + bx + c = 0$. Then, you can plug the values of $a$, $b$, and $c$ into the quadratic formula to solve for $x$.

Q: What is the difference between an exponential function and a quadratic function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. A quadratic function is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: Can I use the quadratic formula to solve any type of equation?

A: No, the quadratic formula can only be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It cannot be used to solve exponential equations or other types of equations.

Q: How do I know if an equation is an exponential equation or a quadratic equation?

A: To determine if an equation is an exponential equation or a quadratic equation, you need to look at the form of the equation. If the equation involves an exponential function, such as $a^x$, it is an exponential equation. If the equation involves a quadratic function, such as $ax^2 + bx + c$, it is a quadratic equation.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, it is often more helpful to use algebraic manipulations and properties of exponential functions to solve the equation, as this can give you a deeper understanding of the solution.

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

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

• Not isolating the variable $x$ correctly
• Not using the correct properties of exponential functions
• Not checking the solution to make sure it is valid
• Not using a calculator to check the solution

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and make sure it is true. You can also use a calculator to check the solution.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial investments
• Modeling electrical circuits

Q: Can I use exponential equations to model any type of real-world phenomenon?

A: Yes, exponential equations can be used to model many types of real-world phenomena, including population growth, chemical reactions, financial investments, and electrical circuits. However, the specific type of equation and the values of the constants will depend on the particular phenomenon being modeled.