To solve the equation \( q^2 - 62 = 38 \), start by isolating \( q^2 \):
\[ q^2 - 62 + 62 = 38 + 62 \]
This simplifies to:
\[ q^2 = 100 \]
Next, take the square root of both sides:
\[ q = \pm \sqrt{100} \]
Since \( \sqrt{100} = 10 \), we have:
\[ q = 10 \quad \text{or} \quad q = -10 \]
Thus, the solutions for \( q \) are:
\[ \boxed{10} \quad \text{and} \quad \boxed{-10} \]