Question

If one root of the quadratic equation ax2+bx+c = 0 is double the otherthen show that 2b2=9ac

Answer 1

We have the following

Let α,β are the two zeros of the polynomial ax²+bx+c
Then, α+β=-b/a and αβ=c/a
By the given condition, β=2α
∴, α+2α=-b/a
or, 3α=-b/a
or, α=-b/3a -------(1)  and
α.2α=c/a
or, 2α²=c/a
or, α²=c/2a
or, (-b/3a)²=c/2a [using (1)]
or, b²/9a²=c/2a
or, b²/9a=c/2
or, 2b²=9ac (Proved).