Let

We need to find the roots of Δ = 0.

Recall that the value of a determinant remains same if we apply the operation R_i→ R_i + kR_j or C_i→ C_i + kC_j.

Applying C₁→ C₁ + C₂, we get

Applying C₁→ C₁ + C₃, we get

Taking the term (3x + a) common from C₁, we get

Applying R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

Expanding the determinant along C₁, we have

Δ = (3x + a)(1)[(a)(a) – (0)(0)]

⇒ Δ = (3x + a)(a)(a)

∴ Δ = a²(3x + a)

The given equation is Δ = 0.

⇒ a²(3x + a) = 0

However, a ≠ 0 according to the given condition.

⇒ 3x + a = 0

⇒ 3x = –a

Thus, is the root of the given determinant equation.