Please note that a quadratic equation has the form ax²+bx+c=0
Δ refers to the discriminant which is under the squareroot sign in the quadratic equation which is b²-4ac
Case 1. Δ≥0, the roots are real
1.1 Δ>0, there are 2 real solutions
This means b²>4ac. We can simply give x²+3x+2. That is (x+1)(x+2)
1.2 Δ=0, there is only one root
This would mean b²=4ac. We can simply give (x+1)² or x²+2x+1
For Case 1.2, it is like this since if we check the quadratic formula:
-b ± √0 = -b
Case 2. Δ<0, there are no real solutions.
Since we would need to get the squareroot of a negative value, which is imaginary.
This would mean b²<4ac. To give an easy example we let a=b=c=1 so x²+x+1.