The general form of a quadratic (second degree) equation is
[tex]a x^{2} +bx+c=0[/tex], where
[tex]D= b^{2}-4ac [/tex] 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, [tex] x^{2} -5x+7=0[/tex], 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.
