Question

The number of values of a for which the equations x^3+ ax+1= 0 and
x ^4+ ax^2 +1=0 have a common root is?​

Answer 1

x^3+ax+1 = 0…………….(1)

x^4+ax^2+1 = 0…………….(2)

x(x^3+ax)+1 = 0 , put x^3+ax = -1 from eq. (1).

x(-1)+1 = 0 => x = 1.

on putting x=1 in eq.(1). or in eq.(2).

1+a+1=0 => a = -2 . On putting a = -2 in eq.(2).

x^4 -2 x ^2 +1 = 0

or (x^2–1)^2 = 0

or x^2–1 =0 => x= +/- 1

x= +1 satisfies the both given equations and x =-1 does not satify the eq.(1).

Hence x = 1 is a common root .

Hope this helps!