Suppose a, b are integers and a+b is a root of x^2+ax+b=0 . What is...

If " $a+b$ " is a root it satisfies the equation Hence $(a+b)^2+a(a+b)+b=0$ $\Rightarrow 2 \mathrm{a}^2+3 \mathrm{ba}+$\left(\mathrm{b}^2+\mathrm{b}\right)=0$Now since " a " is an integer Discriminant is a perfect square$$\begin{aligned} & \Rightarrow 9 \mathrm{~b}^2-8\left(\mathrm{~b}^2+\mathrm{b}\right)=\mathrm{p}^2 \text { for same } \mathrm{p} \in \mathrm{Z} \\ & \Rightarrow (\mathrm{~b}-4)^2-16=\mathrm{p}^2 \\ & \Rightarrow (\mathrm{~b}-4+\mathrm{b})(\mathrm{b}-4-\mathrm{p})=16 \\ & \mathrm{~b}-4+\mathrm{p}= \pm 8, \mathrm{~b}-4-\mathrm{p}= \pm 2, \mathrm{~b}-4+\mathrm{p}=\mathrm{b}-4-\mathrm{p}= \pm 4 \end{aligned}$$So,$b-4=5,-5,4,-4 \Rightarrow \mathrm{b}=9,-1,8,0 \Rightarrow \left(\mathrm{~b}^2\right)_{\max }=81