Question

If p, q, r are in A.P. and are positive, the roots of the quadratic

equation p $x^2$ + qx + r = 0 are all real for ___.

• A.
• B.
• C. All p and r
• D. No p and r

Answer 1

The correct option is A

p,q,r are positive and are in A.P.

∴ q = $\frac{p+r}{2}$ .........(i)

∵ The roots of p $x^2$ + qx + r = 0 are real

⇒$q^2$ ≥ 4pr ⇒ ${\left[\frac{p+r}{2}\right]}^2$ ≥ 4pr [using (i)]

⇒$p^2$ + $r^2$ - 14pr ≥ 0

⇒ ${\left(\frac{r}{p}\right)}^2$ - 14 ($\frac{r}{p}$) + 1 ≥ 0 ( ∵ p>0 and p ≠ 0)

${(\frac{r}{p}-7)}^2$ - $({4\sqrt{3})}^2$ ≥ 0

⇒| $\frac{r}{p}$ - 7| ≥ 4 $\sqrt{3}$.