Question

If x = – 4 is a root of Δ = `|(x, 2, 3),(1, x, 1),(3, 2, x)|` = 0, then find the other two roots.

Answer 1

Applying R₁ → (R₁ + R₂ + R₃), we get

`|(x + 4, x + 4, x + 4),(1, x, 1),(3, 2, x)|`

Taking (x + 4) common from R₁, we get

Δ = `(x + 4) |(1, 1, 1),(1, x, 1),(3, 2, x)|`

Applying C₂ → C₂ – C₁, C₃ → C₃ – C₁ , we get

Δ = `(x + 4)|(1, 0, 0),(1, x - 1, 0),(3, -1, x - 3)|`

Expanding along R₁,

∆ = (x + 4)[(x – 1)(x – 3) – 0].

Thus, ∆ = 0 implies x = – 4, 1, 3.