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Q.
If roots of the cubic $64 x^3-144 x^2+92 x-15=0$ are in arithmetic progression, then the difference between the largest and smallest root is equal to
Sequences and Series
Solution:
Let the roots of the cubic in A.P. be $a - d , a , a + d$.
Now, sum of the roots $=3 a =\frac{144}{64}=\frac{9}{4} \Rightarrow a =\frac{3}{4}$
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 $=2 d =1$