Question

If the roots of the equation $x^3+3p{x}^2+3qx+r=0$ are in A.P, then the condition is:

• A. $2p^3=3pq+r$
• B. $2p^3=3pq$
• C. $2p^3+r=3pq$
• D. $2p^3-r=-3pq$

Answer 1

The correct option is C $2p^3+r=3pq$

As roots are in A.P. then
Let $a-d,a,a+d$ are the roots of $x^3+3p{x}^2+3qx+r=0$
$S_1=-3p\Rightarrow a=p$
Substituting this in equation we get
${(-p)}^3+3p{(-p)}^2+3q(-p)+r=0\Rightarrow -p^3+3p^3-3pq+r=0\Rightarrow 2p^3+r=3pq$