Animated Solution for Mathematics - Quadratic Equations: Find number of real roots of equation $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is

The given equation is:

This is a quartic equation disguised in exponential form. To simplify the landscape, we let $t=e^x$.

Since $e^x$ is always positive, we must respect the domain constraint $t>0$. The equation transforms into the following polynomial:

This is a reciprocal equation because the coefficients $(1,1,-4,1,1)$ mirror each other perfectly.

To exploit this symmetry, we divide the entire equation by $t^2$. Since $t>0$, we know $t^2 eq 0$, making this operation valid:

Next, we group the terms by their powers:

Let $u=t+\frac{1}{t}$. Squaring this expression yields:

Substituting this into our grouped equation, the quartic expression reduces to a quadratic:

Factoring the quadratic gives: This provides two potential candidates: $u=-3$ and $u=2$.

We apply the AM-GM inequality to the substitution $u=t+\frac{1}{t}$. For any $t>0$:

The candidate $u=-3$ fails this condition and is therefore rejected. We proceed with $u=2$:

This yields $t=1$. Back-substituting $e^x=t$, we get $e^x=1$, which implies: $x=0$

We have successfully navigated the fortress to find the single, elegant solution.