Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 0, then the roots of the equation (x – a) (x – β) + c = 0 are :

Question

Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 0, then the roots of the equation (x – α) (x – β) + c = 0 are :

(a) a, c 

(b) b, c 

(c) a, b 

(d) a + c, b + c

Answer 1

(c) a, b

The given equation is (x – a) (x – b) = c 

⇒ x² – (a + b)x + (ab – c) = 0 

As α, β are the roots of this equation, so 

α + β = a + b and αβ = ab – c 

Let γ , δ be the roots of the equation (x – α) (x – β) + c = 0 

i.e., γ , δ are the roots of the equation x² – (α + β)x + (αβ + c)= 0

∴ γ + δ = α + β = a + b                                 ...(i) 

γδ = αβ + c = – c + c = ab                           ...(ii)

∴ From (i) and (ii) we can infer that the roots of the equation 

(x – α) (x – β) + c = 0 are a and b.