Roots of an equation - Exponential

Question

Find the number of roots of the equation: $$(9/10)^x = x - x^2 -3$$

I attempted the question by taking log and then using the derivative to determine the critical points - since the function is clearly positive at x=1.

Is there a shorter/easier way of doing this ? Does Descartes Rule of Signs work here ?

Thanks a lot in advance !

Answer 1

$$\bigg(\frac{9}{10}\bigg)^x=x-x^2-3=-\bigg[\bigg(x-\frac{1}{2}\bigg)^2+3-\frac{1}{4}\bigg]$$

So $$\underbrace{\bigg(\frac{9}{10}\bigg)^x}_{\text{>0}}=\underbrace{-\bigg(x-\frac{1}{2}\bigg)^2-\frac{11}{4}}_{\text{<0}}.$$

So Equation has no real Solution.

Graph:

Answer 2

$$x-x^2-3$$ is always negative (even $\le-\dfrac{11}4$), hence no solution.