The discriminant (#Delta#) tells us something about the roots (solutions) of a quadratic equation without us having to solve the equation first.

#Delta = b^2-4ac" where " ax^2 +bx+c=0#

From #" " 2x^2+3x+1=0#

#Delta = 3^2 -4(2)(1) = 9-8 =1#

#Delta =1#

What does this tell us?

If #Delta <0 " "rarr# the roots are non-real (imaginary)

If #Delta>= 0 " "rarr# the roots are Real. (they do exist!)

If #Delta = 0 " "rarr# there are 2 equal roots (ie one answer)

If #Delta > 0 " "rarr# there are 2 distinct Real roots. (different)

If #Delta "is a square "rarr# the roots are Rational.

If #Delta "is not a square "rarr# the roots are Irrational.

#1# is a perfect square, so the Roots are Real, distinct and Rational.