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If one root of the quadratic equation ax 2 + bx + c = 0 is equal to the n – th power of the other, then show that (ac n ) 1/n + 1 + (a n c) 1/n + 1 + b = 0

 If one root of the quadratic equation ax2 + bx + c = 0 is equal to the n – th  power of the other,  then show that
(acn)1/n + 1 + (an c)1/n + 1 + b = 0

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
9 years ago
Sol. Let α, β be the roots of eq. ax2 + bx + c = 0
According to the question,
Β = αn
Also α + β = -b/a ; αβ = c/a
α β = c/a ⇒ α. α n = c/a ⇒ α = (c/a)1/n + 1
then α + β = - b/a ⇒ α + αn = -b/a
or (c/a)1/n + 1 + (c/a)n/n + 1 = -b/a
⇒ a .(c/a)1/n + 1 + a .(c/a)n/n + 1 + b = 0
⇒ an/n + 1 c1/n + 1 + a1/n + 1 c n/n + 1 + b = 0
⇒ (an c) 1/n + 1 + (can) 1/n + 1 + b = 0
Hence proved.

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