In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if X X is a quasi-complete intersection curve in P 3 \mathbb P^3 , then I X I_X has a minimal free resolution \[ 0 → ⊕ i = 1 μ − 3 S ( d i + 3 + c 1 ) → ⊕ i = 1 2 μ − 4 S ( − e i ) → ⊕ i = 1 μ S ( − d i ) → I X → 0 , 0\to \oplus _{i=1}^{\mu -3} S(d_{i+3}+c_1)\to \oplus _{i=1}^{2\mu -4}S(-e_i)\to \oplus _{i=1}^\mu S(-d_i)\to I_X\to 0, \] where d i , e i ∈ Z d_i,e_i\in \mathbb Z and c 1 = − d 1 − d 2 − d 3 c_1=-d_1-d_2-d_3 . Therefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of I X I_X . Also we give a relation for the degrees of syzygy modules of I X I_X . Using this theorem, one can construct a smooth quasi-complete intersection curve X X such that the number of minimal generators of I X I_X is t t for any given positive integer t ∈ Z + t\in \mathbb Z^+ .