We are tasked with solving the equation:

\[
e^{2x} - e^x = 2
\]

Here is a step-by-step breakdown of how to solve this equation.

### Step 1: Substitution
Let’s make a substitution to simplify the equation. Define:

\[
y = e^x
\]

Now, since \( e^{2x} = (e^x)^2 = y^2 \), we can rewrite the equation as:

\[
y^2 - y = 2
\]

### Step 2: Form a quadratic equation
Rearrange the equation into a standard quadratic form:

\[
y^2 - y - 2 = 0
\]

### Step 3: Solve the quadratic equation
Solve the quadratic equation using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( y^2 - y - 2 = 0 \), we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -2 \)

Substitute these values into the quadratic formula:

\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\]
\[
y = \frac{1 \pm \sqrt{1 + 8}}{2}
\]
\[
y = \frac{1 \pm \sqrt{9}}{2}
\]
\[
y = \frac{1 \pm 3}{2}
\]

### Step 4: Calculate the solutions for \( y \)
Now, solve for the two possible values of \( y \):

\[
y_1 = \frac{1 + 3}{2} = \frac{4}{2} = 2
\]
\[
y_2 = \frac{1 - 3}{2} = \frac{-2}{2} = -1
\]

### Step 5: Solve for \( x \)
Now recall that we substituted \( y = e^x \). We now solve for \( x \) by considering each value of \( y \).

#### Case 1: \( y = 2 \)
We have \( e^x = 2 \). Take the natural logarithm of both sides:

\[
x = \ln(2)
\]

#### Case 2: \( y = -1 \)
We have \( e^x = -1 \). However, this case has no real solution because the exponential function \( e^x \) is always positive for real values of \( x \), and it cannot be equal to a negative number. Therefore, there is no real solution for this case.

### Step 6: Final Solution
The only real solution is:

\[
x = \ln(2)
\]

---

Would you like further details on any step or have any questions?

Here are 5 related questions to extend your learning:

1. How would you solve \( e^{3x} - e^x = 6 \)?
2. What is the general method for solving exponential equations?
3. How can you use the properties of logarithms to simplify exponential equations?
4. Can you solve \( 2e^x - 5e^{-x} = 3 \)?
5. How do you handle exponential equations when dealing with complex numbers?

**Tip:** When solving quadratic equations, the quadratic formula is a powerful tool, but always check if factoring can make the process quicker!

Discover how to solve the exponential equation e^2x - e^x = 2 with a detailed step-by-step breakdown. This solution covers key mathematical concepts, such as exponential equations, quadratic formulas, and logarithms, making it ideal for students in Grades 9-12.

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