An algebra problem by Raffaele Piccirillo

The three roots can be represented by $a-h,a,a+h$ , since the arithmetic mean of the roots must equal one of the roots. I then use Vieta's formulas to form a system of equations. $a=\frac{3+t}{3}$ $a(a+h)+a(a-h)+(a-h)(a+h)=2+3t$ $a(a+h)(a-h)=2t$ The second equation can be rearranged as $(a+h)(a-h)=2+3t-2a^2$ , which can then be substituted into the third.Then use the first equation to get an equation only in $t$ : $(\frac{t+3}{3})(2+3t-2(\frac{t+3}{3})^2)=2t\Rightarrow t(2t^2-9t+9)=0\Rightarrow t=0,\frac{3}{2},3$