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 ______.

Answer 1

roots $\Rightarrow a-d,a,a+d$

$a-d+a+a+d=\frac{144}{64}$

$3a=\frac{144}{64}=\frac{9}{4}$

$a=3/4$

$(a-d)\left(a\right)(a+d)=\frac{15}{64}$

$\frac{3}{4}(a^2-d^2)=\frac{15}{64}$

$\frac{3}{4}\left[\right(\frac{3}{4}{)}^2-d^2]=\frac{15}{64}$

$\frac{9}{16}-d^2=\frac{5}{16}\Rightarrow d^2=\frac{4}{16}=\frac{1}{4}$

$d=1/2$

difference $\Rightarrow a+d-a+d$

$\Rightarrow 2d$

$=2\times \frac{1}{2}=1$