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 $b x^{2}-2 a x+c=0$
A. has no real roots
B. has two equal real roots
C. has two distinct real roots with the same sign
D. has two real roots with opposite signs
Solution
Since $|b-c|(p+q)^{2}-(p-q)^{2} \Rightarrow 4 b c>4 p q$, then $\Delta=4 a^{2}-4 b c<4 p q-4 p q=0 \Rightarrow$ the equation $b x^{2}-2 a x+c=0$ has no real roots. Therefore, the answer is $A$.