Question

can someone define discriminant for me please? i googled it and still don't comprehend it

Answer 1

the discriminant comes for the general quadratic formula and is

\[\Delta = b^2 - 4ac\]

its used to determine the type of roots or zeros, that a quadratic equation can have

here is a nice explanation

http://www.regentsprep.org/Regents/math/algtrig/ATE3/discriminant.htm

Answer 2

In the case of a quadratic equation of the form \[ax^2+bx+c = 0, \,a\ne0\]the discriminant is the expression \(b^2-4ac\)

There are 3 cases:
\[b^2-4ac=0\]Perfect square, there is one positive real root of multiplicity 2 (if you don't know about multiplicity, it doesn't matter)
\[b^2-4ac > 0\]two real roots
\[b^2-4ac < 0\]two complex roots in the form \(a\pm bi\), also known as a complex conjugate pair

Answer 3

The discriminant appears under the radical sign in the formula for the solution to the quadratic equation:
\[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]and thus drives the nature of the solutions. If it is 0, then the solution is just \(x=-\dfrac{b}{2a}\) (which you might recognize as being the x-coordinate of the vertex).

If the discriminant is positive, then we will have two different values for \(x\), both real.

If the discriminant is negative, we'll have the square root of a negative number as part of the expression, giving us two complex roots.