## Task 28/77

Provide a necessary and sufficient condition for the quadratic function $f(x)=a x^{2}+b x+c$ with real coefficients $a ; b ; c(a \neq 0)$ such that for exactly one real number $x_{0}$, it holds that $f\left(x_{0}\right)=x_{0}$.
The problem statement is equivalent to the question under which condition the equation $x_{0}=$ $a x_{0}^{2}+b x_{0}+c$ has exactly one solution. Equivalent transformation yields

$$
x_{0}^{2}+\frac{b-1}{a} x_{0}+\frac{c}{a}=0 \quad ; \quad D=\left(\frac{b-1}{2 a}\right)^{2}-\frac{c}{a}=0
$$

thus $(b-1)^{2}=4 a c$.

Since all inferences are also valid in the reverse direction, this condition is both necessary and sufficient. It is easily verified that $x_{0}=\frac{1-b}{2 a}$ results.