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Using a soft-wall AdS/QCD approach we derive the Schrödinger-type equation of motion for the tetraquark wave function, which is dual to the dimension-4 AdS bulk profile. The latter coincides with the number of constituents in the leading Fock state of the tetraquark. The obtained equation of motion is solved analytically, providing predictions for both the tetraquark wave function and its mass. A low mass limit for possible tetraquark states is given by M≥2κ=1  GeV, where κ=0.5  GeV is the typical value of the scale parameter in soft-wall AdS/QCD. We confirm results of the COMPASS Collaboration recently reported on the discovery of the a1(1414) state, interpreted as a tetraquark state composed of light quarks and having JPC=1++. Our prediction for the mass of this state, Ma1=√2  GeV≃1.414  GeV, is in good agreement with the COMPASS result Ma1=1.414+0.015−0.013  GeV. Next we included finite quark mass effects, which are essential for the tetraquark states involving heavy quarks.