Question

If the difference in the roots of the equation x² – px + q = 0 is unity, then which one of the following is correct ?

(a) p² + 4q = 1 

(b) p² – 4q = 1 

(c) p² + 4q = – 1 

(d) p² – 4q = – 1

Answer 1

(b) p² - 4q = 1. 

Given, x² – px + q = 0 

Let α, β be the roots of the given equation. Then, 

α + β = – \(\frac{(-p)}{1}\) = p             ...(i),                αβ = \(\frac{q}{1}\) = q                        ...(ii) 

Also, α - β = 1 (given)                                     ...(iii) 

∴ From (i) and (iii), 2α = p + 1 ⇒ α = \(\frac{p+1}{2}\)

∴ From (i) and (iii), 2β = p – 1 ⇒ b = \(\frac{p-1}{2}\) 

Substituting these values of a and b in (ii), we have \(\big(\frac{p+1}{2}\big)\)\(\big(\frac{p-1}{2}\big)\) = q

⇒ \(\frac{p^2-1}{4}\) = q ⇒ p² - 1 = 4q ⇒ p² - 4q = 1.