If (D_1=64), (D_2=15), (D_3=0), and (D_4=-9), which one gives irrational distinct roots?

Correct answer: (D_2=15)

The direct answer is A: D2=15 gives irrational distinct roots. For a quadratic equation, D>0 gives two real distinct roots. If D is a perfect square, the roots can be rational when the coefficients are rational; if D is not a perfect square, the roots are irrational. Check each value: D1=64 is positive but 64 is a perfect square, so it does not give irrational roots. D2=15 is positive and 15 is not a perfect square, so it gives real irrational distinct roots. D3=0 gives equal real roots. D4=-9 gives no real roots. Therefore option A is correct. The safest exam chain is: first check positive, then check whether the discriminant is a perfect square.