Abstract. In this paper, we study a spatially homogeneous and anisotropic Bianchi type-III cosmological model sourced by a perfect fluid obeying a quadratic equation of state (QEoS), p = p₁ + p₂ρ + p₃ρ², within the framework of f(R, ℒ_m) gravity proposed by Harko [8]. The non-minimal coupling between the Ricci scalar R and the matter Lagrangian ℒ_m, taken in the specific form f(R, ℒ_m) = R/2 + ℒ_m^α + ϕ, modifies the standard conservation equations and introduces an extra force on massive particles, thereby offering a purely geometric route to dark energy without invoking exotic fields [10,14].
The f(R, ℒ_m) field equations are derived for the Bianchi type-III line element and exact closed-form solutions for the three directional scale factors are obtained by invoking the proportionality condition between the shear scalar and the scalar expansion [24]. Using the QEoS, explicit expressions are derived for the energy density ρ, pressure p, mean Hubble parameter H, scalar expansion θ, and deceleration parameter q. A key result is that q = (δ + 1)/(2δ - 1) is a constant determined solely by the anisotropy parameter δ, which reproduces the observationally favoured value q₀ ≈ -0.55 [3] for δ ≈ 1.56, signalling a transition from early deceleration to late-time accelerated expansion consistent with Type Ia supernovae [1,2] and BAO observations [4].
The equation of state (EoS) parameter ω = p/ρ is analysed and found to evolve from near-matter values at early epochs toward the dark energy regime ω < -1/3 at late times, admitting both quintessence and phantom behaviour a feature facilitated by the quadratic EoS and consistent with recent observational constraints on f(R, ℒ_m) models [11,12,15]. The squared adiabatic sound speed C_s² = p₂ + 2p₃ρ is computed and shown to satisfy 0 ≦ C_s² ≦ 1 throughout the cosmic evolution, confirming classical stability against small perturbations [3]. The statefinder diagnostic pair {r,s} [21] is evaluated and the resulting trajectory in the r-s plane is distinctly different from ΛCDM, quintessence, and Chaplygin gas models, providing a clear geometric discriminator for future surveys such as DESI and Euclid. Furthermore, the shear scalar decays as σ² ∝ t^-2, demonstrating late-time isotropisation in accord with cosmological observations [20].
Our results establish that the Bianchi type-III cosmological model with a quadratic EoS in f(R, ℒ_m) gravity is a physically viable, classically stable, and observationally consistent alternative to the standard ΛCDM cosmology for describing the late-time accelerated universe.

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