Question

The number of real solutions of the equation \(1 + |e^x - 1| = e^x(e^x - 2)\) is

• A. 1
• B. 2
• C. 4
• D. 8

Hint:

Check both cases for absolute value equations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Let \(y = e^x - 1\), then solve for \(y\).
Step 2: Detailed Explanation:
RHS = \(e^{2x} - 2e^x = (e^x - 1)^2 - 1 = y^2 - 1\)
LHS = \(1 + |y|\)
Equation: \(1 + |y| = y^2 - 1 \rightarrow |y| = y^2 - 2\)
For \(y \ge 0\): \(y = y^2 - 2 \rightarrow y^2 - y - 2 = 0 \rightarrow y = 2\) or \(-1\) (reject)
For \(y<0\): \(-y = y^2 - 2 \rightarrow y^2 + y - 2 = 0 \rightarrow y = 1\) (reject) or \(y = -2\)
\(y = 2 \rightarrow e^x = 3 \rightarrow x = \ln 3\)
\(y = -2 \rightarrow e^x = -1\) (no solution)
Only one solution.
Step 3: Final Answer:
1 real solution.