Question

If a and b are the zeroes of the quadratic polynomial f(x)=3x^2-7x-6, find a polynomial whose zeroes are (i) a^2 and b^2 (ii) 2a+3b and 3a+2b

Answer 1

For given quadratic f(x)
Sum of roots = a+b=-(-7)/3 = 7/3
( = - coefficient of x /coefficient of x^2)
Product of roots =ab =-6/3 = -2
( = constant term/coefficient of x^2)
(i)
We need a quadratic with roots a^2 and b^2

Here sum of roots =a^2 + b^2
=(a + b) ^2 - 2ab = (7/3)^2 - 2(-2) = 85/9
Product of roots = a^2*b^2 = (ab)^2
=(-2)^2 = 4
Which can be written as 36/9 when multiplied and divided with 9.

From above equations we get coefficients of x2 =9 , x= - 85 and constant term=36.
So equation is 9x^2 - 85x +36

(ii)
Here sum of roots =3a +2b +2a +3b
=5(a+b) = 5*(7/3) = 35/3

​​​​​Product of roots = (2a +3b)(3a +2b)
=6a^2 +9ab+4ab+6b^2=6(a^2 +b^2) +13ab
=6(85/9) +13(-2) = (170 - 78)/3
= 92/3
From above equations we get coefficients of x2 =3 , x= - 35 and constant term=92.
So equation is 3x^2 - 35x +92.

Hope this helps :)