Complete the square for exponents

Question

How does $\exp(2x)-2+\exp(-2x) = (\exp(x) - \exp(-x))^2$ ?

I am having trouble using complete the square.

Answer 1

Just substitute $exp(x)=a$

Write down the first two summands: $a^2-2$

Completing the square

To get the second binomial formula, -2 has to be -2ab. a is $exp(x)$. Thus b has to be $exp(-x)$, because $exp(x) \cdot exp(-x)=1$ . Now it is obvious, that $b^2=exp(-2x)$.

$a^2-2ab+b^2=(a-b)^2 \Rightarrow exp(2x)-2+exp(-2x)= \left( exp(x)-exp(-x) \right)^2$