The general form of a quadratic (second degree) equation is
![a x^{2} +bx+c=0](https://tex.z-dn.net/?f=a%20x%5E%7B2%7D%20%2Bbx%2Bc%3D0)
, where
![D= b^{2}-4ac](https://tex.z-dn.net/?f=D%3D%20b%5E%7B2%7D-4ac%20)
is called the Discriminant.
The Discriminant determines how many roots the equation will have as follows:
i) if D>0, the equation has 2 roots.
ii) if D=0, the equation has 1 double root.
iii) if D<0, the equation has no roots.
In our equation,
![x^{2} -5x+7=0](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20-5x%2B7%3D0)
, a=1, b=-5, c=7
so the discriminant is D=(-5)^2-4*1*7=25-28<0
Thus the equation has no roots.
Remark: the equation has no roots in the Real numbers, but it has 2 roots in a larger set of numbers to be discussed in the future, the Complex numbers.