Question

If roots of the cubic $64x^3-144x^2+92x-15=0$ are in arithmetic progression, then the difference between the largest and smallest root is equal to

• A. 2
• B. 1
• C. $\frac{1}{2}$
• D. $\frac{3}{8}$

Answer 1

Answer: (B)

Let the roots of the cubic in A.P. be $a-d,a,a+d$.
Now, sum of the roots $=3a=\frac{144}{64}=\frac{9}{4}\Rightarrow a=\frac{3}{4}$
Also, product of the roots $=a\left(a^2-d^2\right)=\frac{15}{64}\Rightarrow a\left(a^2-d^2\right)=\frac{3}{4}\left(\frac{9}{16}-d^2\right)$
$\Rightarrow d^2=\frac{9}{16}-\frac{5}{16}\Rightarrow d^2=\frac{4}{16}\Rightarrow d=\pm \frac{1}{2}$
Hence difference between largest and smallest roots $=2d=1$