If one zero of a quadratic polynomial p(x)`=ax^(2)+bx+c` is square of the other, then give the relation in a, b and c.

Answer

N/a

Explanation

Let one zero of `ax^(2)+bx+c` be `alpha`.
`:. ` other zero of `ax^(2)+bx+c` be `alpha^(2)`.
`:." "` sum of zeroes`= alpha+alpha^(2)=-(b)/(a) " "` ...(1)
and product of zeroes `=alpha.alpha^(2)=(c )/(a)impliesalpha^(3)=(c )/(a) " "` ......(2)
Now, from equation (1) since,
`alpha+alpha^(2)=(-b)/(a)`
`:. " "(alpha+alpha^(2))^(3)=(-(b)/(a))^(3)" "` (cubing both sides)
`implies alpha^(3)+(alpha^(2))^(3)+3alpha.alpha^(2)(alpha+alpha^(2))=(-b^(3))/(a^(3)) " " ["using "(a+b)^(3)=a^(3)+b^(3)+3ab(a+b)]`
`implies " " alpha^(3)+(alpha^(3))^(2)+3alpha^(3)(alpha+alpha^(2))=(-b^(3))/(a^(3))`
`implies " " (c )/(a)+((c )/(a))^(2)+3.(c )/(a)(-(b)/(a))=(-b^(3))/(a^(3)) " "` [from (1) and (2)]
Multiplying both sides by `a^(3)`
`a ^(2)c+ac^(2)-3abc=-b^(3)`
`implies " " a^(2)c+ac^(2)+b^(3)=3abc`
This is the required relation. `" "` Ans.