Question

Solve the equation $4x^3-24x^2+23x+18=0$, having given that the roots are in arithmetical progression.

Answer 1

Denote the roots by $a-b,a,a+b$; then the sum of the roots is $3a$; the sum of the product of the roots two at a time is $3a^2-b^2$; and the product of the roots is $a(a^2-b^2)$; hence we have the equations
$3a=6,3a^2-b^2=\frac{23}{4},a(a^2-b^2)=-\frac{9}{2}$;
From the first equation we find $a=2$, and from the second $b=\pm \frac{5}{2}$, and since these values satisfy the third, the three equations are consistent.
Thus the roots are $-\frac{1}{2},2,\frac{9}{2}$.