Let R be a regular semilocal domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let (Rⁿ, q: Rⁿ → R) be a quadratic space over R such that the quadric (q = 0) is smooth over R. If the quadratic space (Rⁿ, q: Rⁿ → R) over R is isotropic over K, then there is a unimodular vector v ∈ Rⁿ such that q(v) = 0. If char(R) = 2, then in the case of even n the assumption on q is equivalent to the fact that q is a nonsingular quadratic space and in the case of odd n > 2 this assumption on q is equivalent to the fact that q is a semiregular quadratic space.