Thus, the number of interior lattice points is $ \boxed{10} $.Question: Compute the sum of the roots of the equation
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- For $n = 6$:
- I = \frac{(6-1)(6-2)}{2} = \frac{5 \cdot 4}{2} = 10
- This formula arises from known results in triangular lattice geometry: the number of interior points in a triangle of side $n$ is $\frac{n(n-2)(n-1)}{6}$, or equivalently $\frac{(n-1)(n-2)}{2}$.
- u\sqrt{u} - 5u + 6\sqrt{u} = 0,
- given that all of the roots are real and nonnegative.
- Solution: Let us make the substitution $ x = \sqrt{u} $, so that $ u = x^2 $. Substituting into the equation gives:
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