2𝑒2𝑥 − 3𝑒𝑥 + 1 = 0

This is a quadratic equation in terms of e^x. We can solve it by using the quadratic formula.

Step 1: Identify a, b, and c in the quadratic equation ax^2 + bx + c = 0. Here, a = 2, b = -3, and c = 1.

Step 2: Substitute a, b, and c into the quadratic formula, which is x = [-b ± sqrt(b^2 - 4ac)] / (2a).

Step 3: Calculate the discriminant (b^2 - 4ac). Here, the discriminant is (-3)^2 - 421 = 9 - 8 = 1.

Step 4: Substitute b, a, and the discriminant into the quadratic formula. Here, x = [3 ± sqrt(1)] / 4.

Step 5: Solve for x. Here, x = 1 or x = 0.5.

Step 6: Since x is actually e^x in the original equation, we have e^x = 1 or e^x = 0.5.

Step 7: Solve for x in each case. Here, x = ln(1) = 0 or x = ln(0.5) = -0.6931.

So, the solutions to the equation 2e^2x - 3e^x + 1 = 0 are x = 0 and x = -0.6931.