Question

If α and β are zeroes of the quadratic polynomial 4x² + 4x + 1, then find quadratic polynomial whose zeroes are 2α and 2β.

Answer 1

Lets divide the quadratic equation by 4 such that we have it in the form x² + (α+β)x + αβ

So after dividing by 4 we have

So we have

α + β = 1       ............(1)

and αβ = 41​    ............(2)

So if zeroes to the new quadratic equation are 2α and 2β, then 

2α + 2β = 2(α+β)

= 2(1) 

= 2     .......... Using (1)

and,

2α × 2β = 4αβ

= 4 × 1​/4

= 1   ......... Using (2)

So the new quadratic equation with its roots 2α and 2β will be

x² + (2α+2β)x + 2α × 2β

Putting the values, the equation would be,

x² + 2x + 1