First, when a=0, negative real roots can be obtained. When a is not equal to 0, if the discriminant 1-4a is greater than or equal to 0, then ax 2+x+1= 0 has at least one real root, so a is less than or equal to 1/4.
It is discussed that when a is greater than 0 and less than or equal to 1/4 and-1/2a is less than 0, the function Y = AX 2+X+ 1 has an upward opening, the symmetry axis x=- 1/2a is on the left side of the y axis, and the curve Y = ax2+ when x=0.
When a is less than 0 and-1/2a is greater than 0, the function y = ax 2+x+1has a downward opening, the symmetry axis x=- 1/2a is on the right side of the y axis, and the value of the curve y = ax 2+x+1is in.
2) ax 2+x+1= 0 has at least one positive real root if and only if-
According to the analysis of the first question, a is less than 0.
3) ax 2+x+1= 0 has two negative real roots if and only if-
According to the first question, the conclusion is that A is greater than 0 and A is less than 1/4. Note that there are two roots here.
4) AX 2+X+1= 0 has at least one positive real root if and only if-
According to the second question, the necessary and sufficient condition is that A is less than 0, so A is less than-1.
5) ax 2+x+1= 0 has at least one negative real root if and only if-
According to the first question, you can fill in a less than 1