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If α, β are the roots of x 2 + px + q = 0 and γ, δ are the roots of x 2 + rx + s = 0, evaluate (α – γ) (α – δ) (β – γ) (β – δ) in terms of p, q, r and s. Deduce the condition that the equations have a common root.

 If α, β are the roots of x2 + px + q = 0 and γ, δ are the roots of x2 + rx + s = 0, evaluate (α – γ) (α – δ) (β – γ) (β – δ) in terms of p, q, r and s.
Deduce the condition that the equations have a common root.
 

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
9 years ago
Sol. ∵ α, β are the roots of x2 + px + q = 0
∴ α + β = -p, αβ = q
∵ γ, δ are the roots of x2 + rx + s = 0
∴ γ + δ = -r, γδ = s
Now, (α – γ) (α – δ) (β – γ) (β – δ)
= [α2 – (γ + δ)α + γδ] [β2 – (γ + δ) β + γδ]
= α2 + rα + s] [β2 + rβ + s] [∵α, β are roots of x2 + px + q = 0
∴ α2 + pα + q = 0 and β2 + pβ + q = 0]
= [(r – p) α + (s – q)] [(r – p) β + (s – q)]
= (r – p)2 α β + (r – p) (s – q) (α + β) + (s – q)2
= q(r – p)2 – p(r – p) (s – q) + (s – q)2
Now if the equations x2 + px + q = 0 and x2 + rx + s = 0 have a common root say α, then α2 + pα + q = 0 and α2 + rα + s = 0
⇒ α2/ps – qr = α/q – s = 1/r - p
⇒ α2 = ps – qr/r – p and α = q – s/r – p
⇒ (q – s)2 = (r – p) (ps – qr) which is the required condition.

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