If `x=(2)/(3)` and x=-3 are zeroes of the quadratic polynomial `ax^(2)+7x+b`, then find the values of a and b.

Explanation

Let `p(x)=ax^(2)+7x+b`
`:' x=(2)/(3)` and x=-3 are zeroes of p(x)
`:. " " p((2)/(3))=0`
`implies " " a((2)/(3))^(2)+7((2)/(3))+b=0`
`implies " " (4a)/(9)+(14)/(3)+b=0`
`implies " " b=-(4a)/(9)-(14)/(3)` ...........................................(1)
and `" " p(-3)=0`
`implies " " a(-3)^(2)+7(-3)+b=0`
`implies " " 9a-21-(4a)/(9)-(14)/(3)=0 " "`[from(1)]
`implies " " (81a-4a)/(9)=(14)/(3)+21=(14+63)/(3)`
`implies " " (77a)/(9)=(77)/(3)`
`implies " "` a=3
From equation(1)
`b=(-4xx3)/(9)-(-14)/(3)=-6`
a=3, b=-6 `" " ` Ans.