Let [tex]y=ax^2+bx+c[/tex] be a quadratic equation; a is not equal to 0.
The discriminant of [tex]ax^2+bx+c[/tex] is defined as [tex]D=b^2-4ac[/tex].
The number of (real number) solutions of a quadratic equation is determined by the discriminant as follows:
If D>0, then the equation has 2 (different) solutions.
If D=0, the equation has 1 real solution.
If D<0, the equation has no real solutions.
Thus, we calculate the discriminants in each equation, and decide on the number of solutions accordingly:
i) a=-3, b=1, c=12; D=(-3)^2-4*1*12=9-48<0 (no real solutions).
ii) a=2, b=-6, c=5; D=2^2-4*(-6)*5=4+120>0 (2 real solutions).
iii) a=1, b=7, c=-11; D=1^2-4*7*(-11)=1+28*11>0 (2 real solutions).
iv) a=-1, b=-8, c=-12; D=(-1)^2-4(-8)(-12)=1-48*12<0 (no real solutions).
Answer: no solution, 2 solutions, 2 solutions, no solution.
