Mathematics

The key is to connect the difference between roots to the sum and product you can read straight from the quadratic. If you call the roots $r$ and $s$, you know their sum is $\lambda$ and their product is $8$ (from Vieta's formulas). The difference $|r - s| = 2$ gives you another equation—square it to get $(r-s)^2 = 4$, then expand using $(r-s)^2 = (r+s)^2 - 4rs$. This lets you solve for $\lambda$ without finding the roots explicitly.

The key is to connect the difference between roots to the sum and product you can read straight from the quadratic. If you call the roots $r$ and $s$, you know their sum is $\lambda$ and their product is $8$ (from Vieta's formulas). The difference $|r - s| = 2$ gives you another equation—square it to get $(r-s)^2 = 4$, then expand using $(r-s)^2 = (r+s)^2 - 4rs$. This lets you solve for $\lambda$ without finding the roots explicitly.