Solve the equation:
\[ e^{2x} + 2e^{x} - 3 = 0 \]
Answer (2)
e 2 x + 2 e x − 3 = 0 ( ∗ ) ( e x ) 2 + 2 e x ⋅ 1 + 1 2 − 1 2 − 3 = 0 ( e x + 1 ) 2 − 1 − 3 = 0 ( e x + 1 ) 2 − 4 = 0 ( e x + 1 ) 2 = 4 ⟺ e x + 1 = 4 → e x + 1 = 2 e x = 2 − 1 e x = 1 e x = e 0 ⟺ x = 0
To solve the equation e 2 x + 2 e x − 3 = 0 , we first substitute y = e x , leading to the quadratic y 2 + 2 y − 3 = 0 . Factoring gives ( y + 3 ) ( y − 1 ) = 0 , leading to y = 1 , and thus the solution is x = 0 .
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