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Q.
If $p, q, r$ are positive and are in A.P., the roots of quadratic equation $p x^{2}+q x+r=0$ are all real for
Sequences and Series
Solution:
Since $p, q, r$ are in A.P. so $2 q=p+r$.
The roots of the equation $p x^{2}+q x+r=0$ are real if and only if
$q^{2}-4 p r \geq 0 $
$\Rightarrow \left(\frac{p+r}{2}\right)^{2}-4 p r \geq 0$
$\Rightarrow p^{2}+r^{2}-14 p r \geq 0 $
$\Rightarrow \frac{p^{2}}{r^{2}}-14 \frac{p}{r}+1 \geq 0$
$\Rightarrow \left(\frac{p}{r}-7\right)^{2}-48 \geq 0 $
$\Rightarrow \left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$