Quadratic EquationMedium
Question
If x4 + ax3 + bx2 + cx + d = 0 has four positive real roots, then-
Options
A.ac - 16d ≥ 0
B.ac - 16d ≤ 0
C.b2 - 36d ≥ 0
D.b2 - 36d ≤ 0
Solution
Let α1, α2, α3, α4 are the positive real roots
⇒ ∑ α1 = - z, ∑α1α2 = b, ∑α1α2α3 = - c & α1α2α3α4 = d(>0)
By AM ≥ GM
≥ (α1α2α3α4)1/4 ⇒ - 
≥ d1/4 ........ (i)
&
≥ (α1α2α3α4)3/4 ⇒ - 
≥ d3/4 .....(2)
from (1) & (2) ac - 16 d ≥ 0
Now again
≥ ((α1α2α3α4)3/6
⇒
≥d1/2 ⇒ b2 - 36d ≥ 0
⇒ ∑ α1 = - z, ∑α1α2 = b, ∑α1α2α3 = - c & α1α2α3α4 = d(>0)
By AM ≥ GM
≥ (α1α2α3α4)1/4 ⇒ - ≥ d1/4 ........ (i)&
≥ (α1α2α3α4)3/4 ⇒ - ≥ d3/4 .....(2)from (1) & (2) ac - 16 d ≥ 0
Now again
≥ ((α1α2α3α4)3/6⇒
≥d1/2 ⇒ b2 - 36d ≥ 0Create a free account to view solution
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