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 our assumption on q is equivalent to the fact that q is a nonsingular quadratic space and in the case of odd n>2 our assumption on q is equivalent to the fact that q is a semiregular quadratic space.