Question

The number of integers of k for wich the equation x^3-27x+k=0

Answer 1

let a, b the two roots of given equation x^3–27x+k=0.
a^3–27a+k=0……………..(1)
b^3–27b+k=0………………( 2 ) subtract eq.(2)from(1)
a^3-b^3–27(a-b)=0
(a-b)(a^2+ab+b^2)-27(a-b)=0
(a-b)(a^2+ab+b^2–27)=0
Either a-b=0=> a=b
or a^2+ab+b^2–27=0 , put a=b
b^2+b^2+b^2–27=0
3b^2=27
b^2=9
b=+3 , - 3
Put b=+3 in eq.(2)
(+3)^3–27×(+3)+k=0
27–81+k=0
-54+k=0
k=54 ,
i hope this will help you