If (x^2-2\ell x+\ell^2-9=0), what is the nature of roots?

Correct answer: Always real and distinct

The direct answer is A: the roots are always real and distinct, assuming ℓ is real. For the quadratic x^2−2ℓx+ℓ^2−9=0, the coefficients are a=1, b=−2ℓ, and c=ℓ^2−9. The discriminant is D=b^2−4ac = (−2ℓ)^2−4(1)(ℓ^2−9) = 4ℓ^2−4ℓ^2+36 = 36. Since 36 is positive for every real value of ℓ, the equation always has two real and unequal roots. In fact, rewriting gives (x−ℓ)^2=9, so x=ℓ+3 or x=ℓ−3; these differ by 6. Option A is correct. Option B is wrong because equal roots require D=0, not D=36. Option C is wrong because non-real roots require D<0. Option D is wrong because the roots are real for every real ℓ, not only when ℓ=3; ℓ=3 is not a special requirement. The condition that ℓ be real is understood in this school context. Memory cue: positive discriminant means real and distinct roots.