What is the nature of the roots of x²−2x−3=0?

Correct answer: Real and distinct

For a quadratic ax²+bx+c=0, the discriminant D=b²−4ac determines the nature of its roots. In x²−2x−3=0, the coefficients are a=1, b=−2, and c=−3. Thus D=(−2)²−4(1)(−3)=4+12=16, which is positive. A positive discriminant means two real and distinct roots. The result can also be checked by factorisation: x²−2x−3=(x−3)(x+1), so the roots are 3 and −1. They are real and unequal, and one is positive. Equal roots require D=0, while non-real roots require D<0. Therefore option A is correct and option D is contradicted by the actual roots.