No. of real solutions of the equation $\big(\!\frac{9}{10}\!\big)^x = - 3 + x - x^2$

Question

How many real solutions are there of the equation $\left(\dfrac{9}{10}\right)^x=-3+x-x^2$ ? Please illustrate.

Answer 1

Note that $$f(x) = -x^2+x-3 \lt 0 \forall x \in \Bbb{R}$$ ($\Delta = 1-4\cdot3=-11\lt0$; coefficient of the term of max degree is $\lt 0 \Rightarrow $ no intersections with $x-$axis) while $$g(x) = \left(\frac{9}{10}\right)^x \gt 0 \forall x \in \Bbb{R} \Rightarrow $$ there are no real solutions of the equation $$\left(\dfrac{9}{10}\right)^x = - 3 + x - x^2$$