Example 7 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
(2000, National High School Mathematics Competition)
Analysis: Taking $p=1, a=2, q=4, b=2, c=3$, then from the quadratic equation $2 x^{2}-4 x+3=0$ we easily get the answer as $(\mathrm{A})$.