Question

Let α, β be the roots of x^2 - x + p = 0 and γ, δ be the roots . of x^2 - 4x + q = 0.

Answer 1

First quadratic equation is x2-x+p=0,a and b are the roots of this eqn,so a+b=1,a*b=p,b=1-a
Again for second quadratic x2-4x+q,C and d are the roots of the equation,so c+d=5 c*d=q,d=4-c.
So if we will look at the equation
b=1-a,d=4-c we can put some values which should be GP,so start from 1,if we will put 1 then b=0,so you can't put 1
Then try -1 we will b=2,
From geometric progression we can say that it's r(common ratio)is -2 and by that if we will find C &d we will get -4&8.
If we will put c&d value in the above equation of c&d then it is satisfied thus whatever we have assumed is right ,so a=-1,b=2,C=-4,d=8
From quadratic we know that p=a*b,p=-2
q=c*d,q=-32