https://file.aiquickdraw.com/mathbot/file/1729703305713oxmnyaw3.jpg

The equation from the image is:

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

We can solve this equation step by step:

### Step 1: Substitution
Let \( y = e^x \). This simplifies the equation because \( e^{2x} = (e^x)^2 = y^2 \). So the equation becomes:

\[
y^2 - y = 2
\]

### Step 2: Rearrange into a quadratic equation
Rearrange the equation to get:

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

### Step 3: Solve the quadratic equation
Now solve the quadratic equation \( y^2 - y - 2 = 0 \) using the quadratic formula:

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

where \( a = 1 \), \( b = -1 \), and \( c = -2 \). Substituting these values:

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

\[
y = \frac{1 \pm \sqrt{1 + 8}}{2}
\]

\[
y = \frac{1 \pm \sqrt{9}}{2}
\]

\[
y = \frac{1 \pm 3}{2}
\]

Thus, the two solutions are:

\[
y = \frac{1 + 3}{2} = 2 \quad \text{or} \quad y = \frac{1 - 3}{2} = -1
\]

### Step 4: Back-substitute \( y = e^x \)
Now, recall that \( y = e^x \). So:

1. If \( y = 2 \), then \( e^x = 2 \). Taking the natural logarithm of both sides:

   \[
   x = \ln(2)
   \]

2. If \( y = -1 \), then \( e^x = -1 \). However, this has no real solution because the exponential function \( e^x \) is always positive.

### Final solution:
The only valid solution is:

\[
x = \ln(2)
\]

Let me know if you'd like further explanation or if you have any questions! 

Here are 5 related questions you might find useful:
1. How can you solve other types of exponential equations?
2. What is the quadratic formula, and how is it derived?
3. Why does \( e^x \) not have any negative solutions?
4. How can you check if a solution to an equation is valid?
5. How does logarithmic differentiation work?

**Tip:** When dealing with exponential equations, it's often helpful to use substitution to simplify the expression before solving.

Learn how to solve the exponential equation e^(2x) - e^x = 2 by substituting variables to form a quadratic equation. Understand key concepts like logarithms, the quadratic formula, and the properties of exponential functions. This problem is ideal for students in Grades 10-12.

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