1. **State the problem:** Solve for the two possible values of $a$ in the equation $$\frac{a^2}{2} + 27 = 35.$$\n\n2. **Isolate the term with $a^2$:** Subtract 27 from both sides to get $$\frac{a^2}{2} = 35 - 27.$$\n\n3. **Simplify the right side:** $$\frac{a^2}{2} = 8.$$\n\n4. **Eliminate the denominator:** Multiply both sides by 2 to isolate $a^2$: $$\cancel{2} \times \frac{a^2}{\cancel{2}} = 8 \times 2,$$ which simplifies to $$a^2 = 16.$$\n\n5. **Solve for $a$:** Take the square root of both sides: $$a = \pm \sqrt{16}.$$\n\n6. **Calculate the square root:** $$a = \pm 4.$$\n\n**Final answer:** The two possible values of $a$ are $4$ and $-4$.