4. Given positive numbers $p, q, a, b, c$, where $p \neq q$. If $p, a, q$ form a geometric sequence, and $p, b, c, q$ form an arithmetic sequence, then the quadratic equation $b x^{2}-2 a x+c=0(\quad)$.
A. has no real roots
B. has two equal real roots
C. has two distinct real roots of the same sign
D. has two distinct real roots of opposite signs
4. A

From the problem, we know that $p q=a^{2}, 2 b=p+c, 2 c=q+b$. From the last two equations, we get $b=\frac{2 p+q}{3}, c=\frac{p+2 q}{3}$. Therefore, we have
$$
\begin{aligned}
3 c & =\frac{p+p+q}{3} \cdot \frac{p+q+q}{3} \\
& \geqslant \sqrt[3]{p^{2} q} \cdot \sqrt[3]{p q^{2}}=p q=a^{2} .
\end{aligned}
$$

Since $p \neq q$, it follows that $b c>a^{2}$. The discriminant of the equation $\Delta=4 a^{2}-4 b c<0$. Therefore, the equation has no real roots.