The harmonic mean of the roots of the equation is
- A.2
- B.4
- C.6
- D.8
Show answer and solution
Correct option: B
Solution
The harmonic mean of two roots α, β is 2αβ/(α+β).
From Vieta's formulas:
- α + β = (4 + √5)/(5 + √2)
- αβ = (8 + 2√5)/(5 + √2)
HM = 2αβ/(α+β) = 2(8 + 2√5)/(4 + √5) = 2·2(4 + √5)/(4 + √5) = 4.
The key simplification is that 8 + 2√5 = 2(4 + √5), which cancels with the denominator (4 + √5), giving HM = 4.
From Vieta's formulas:
- α + β = (4 + √5)/(5 + √2)
- αβ = (8 + 2√5)/(5 + √2)
HM = 2αβ/(α+β) = 2(8 + 2√5)/(4 + √5) = 2·2(4 + √5)/(4 + √5) = 4.
The key simplification is that 8 + 2√5 = 2(4 + √5), which cancels with the denominator (4 + √5), giving HM = 4.
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