Finding a root approach with a polynomial

Question

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to 0) with a second degree polynomial, find an approach of this root.

Now, my mind went to using Taylor's Theorem. So i've found a second degree polynomial of $e^{-4x}$ with it(it's $8x^2-4x+1$). Now what?

Answer 1

HINT

Now $$e^{-4x} = 5x^2$$ becomes $$5x^2 \approx 8x^2-4x+1$$ which is a quadratic you can solve. The roots should approximate the rotts of the transcendental equation you started with.

Answer 2

For $\;x\;$ close enough to zero we have the good approximation you gave, so

$$8x^2-4x+1=5x^2\iff3x^2-4x+1=0\implies x=\begin{cases}1\\{}\\\frac13\end{cases}$$

Take $\;\frac13\;$ as a good approx. to the wanted zero.