Question

Find the condition that one of the roots of the equation ax^2+bx+c=0 may be positive and the other is negative
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Answer 1

We need distinct real roots, so necessarily Δ=b2−4ac>0. Furthermore, the product of the roots,I.e c/a must be negative. As the sign of c/a is the same as the sign of ac, we have ac<0.

We may note that ac<0 implies Δ>0, so the first condition is redundant, and finally, the equation ax2+bx+c=0 has a positive root and a negative root if and only if ac<0.