Find the integral values of a for which the equation `x^4-(a^2-5a+6)x^2-(a^2-3a+2)=0` has only real roots

Answer

3

Explanation

for the equation
`x^(4)-(a^(2)-5a+6)x^(2)-(a^(2)-3a+2)=0`
to have real root only the equation
`t^(2)-(a^(2)-5a+6)t-(a^(2)-3a+2)=0`
must have both roots greater then or equal to zero
`(a^(2)-5a+6)^(2)+4(a^(2)-3a+2)gt0`
`(a^(2)-5a+6)/(2(a))gt0=a^(2)-5a+6gt0`
`(a-2)(a-3)gt0`
`a in (-infty,2]cup[3,infty)`
`(3)`
`-(a^(2)-3a+2)gt0`
`-(a^(2)-3a+3)gt0`
`-(a-1)(a-2)gt0`
`a in [1,2]`
From 2 & 3 we have integral value of a equal to 1 and 2 which also satisfy condition 1. ltbr `a=1,2`