Question

If the equation `ax^2+x+b=0` and `x^2+bx+a=0` a have common root and the second Equation has equal roots, then `2a^2+b`:

Answer 1

the given equations are `ax^2+x+b=0 & x^2+bx+a = 0`
let `alpha ` be the common root
then`alpha^2/(a-b^2) = -alpha/(a^2-b)= 1/(ab-1)`
`alpha= -(a^2-b)/(ab-1)`
`alpha = (b- a^2)/(ab-1)`eqn(1)
now, `b^2-4(1)(a) = 0`
`b^2= 4a`
sum of roots`=2alpha = -b`
`alpha = -b/2`
equating we get `(b-a^2)/(ab-1) = -b/2`
`2b- 2a^2 = -(ab^2 -b)`
`2b- 2a^2 = -a(4a)+ b`
`2b-2a^2 = -4a^2 +b`
`4a^a-b+2b-2a^2 = 0`
`2a^2+b=0`
option 1