Question

If the equation
(x^4 - 4x^3 + ax^2 + bx + 1) = 0 has only real roots.Then the value of a and b,
a)(6,4)
b)(6,-4)
c)(4,-6)
d)(-4,6)

Solution:

Since it is given that the roots of the equation are real can say x_1,x_2,x_3 and x_4 be the roots of the equation.

So
(x_1 + x_2 + x_3 + x_4) = 4 and (x_1 xx x_2 xx x_3 xx x_4)=1

Now applying AM GM inequality we have

(x_1 + x_2 + x_3 + x_4)/4=(x_1x_2x_3x_4)

This case occurs whenever x_1=x_2=x_3=x_4=1
So,the equation becoems;
(x-1)^4=0

By binomial expansion,we get;
a=6 and b=(-4)