Question
If p, q, r are in A.P. and are positive, the roots of the quadratic
equation p x2 + qx + r = 0 are all real for ___.
equation p x2 + qx + r = 0 are all real for ___.
Answer: Option A
:
A
p,q,r are positive and are in A.P.
∴ q = p+r2 .........(i)
∵ The roots ofp x2 + qx + r = 0 are real
⇒q2≥ 4pr⇒ [p+r2]2≥ 4pr [using (i)]
⇒p2 +r2 - 14pr≥ 0
⇒ (rp)2 - 14 (rp) + 1≥ 0 ( ∵ p>0 and p≠ 0)
(rp−7)2 - (4√3)2≥ 0
⇒| rp - 7|≥ 4 √3.
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:
A
p,q,r are positive and are in A.P.
∴ q = p+r2 .........(i)
∵ The roots ofp x2 + qx + r = 0 are real
⇒q2≥ 4pr⇒ [p+r2]2≥ 4pr [using (i)]
⇒p2 +r2 - 14pr≥ 0
⇒ (rp)2 - 14 (rp) + 1≥ 0 ( ∵ p>0 and p≠ 0)
(rp−7)2 - (4√3)2≥ 0
⇒| rp - 7|≥ 4 √3.
Was this answer helpful ?
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