Quadratics: The Discriminant

Do you remember the quadratic formula? Once you have memorized it, you can use a part of it, called the discriminant, and it helps you quickly visualize a parabola. 

You don't have to memorize the discriminant formula if you know the quadratic formula:

$\dfrac{-b\pm \sqrt{\bf{b^2-4ac}}}{2a}$

The bolded part of the quadratic formula is the discriminant:

$b^2-4ac=D$

So, to find the discriminat of a quadratic equation, put the equation into standard form ($ax^2+bx+c=0$) you can pull out $a$, $b$, and $c$.  Then put it into the discriminant formula to find $D$.

Example: 

$3x^2+2x-8=0$

$\eqalign{b^2-4ac&=D\\2^2-4(3)(-8)&=D\\4+96&=D\\D&=100}$

So, the discriminant of this equation is 100.  

What can we learn from that?

D>0 (positive)	D=0	D<0 (negative)
Parabola has 2 unequal real roots	Parabola has 1 real root	Parabola has two imaginary roots
Example: $\eqalign{x^2+6x-2=0\\\text{Plug into discriminant}\\6^2-4(1)(-2)\\36+8\\44}$ Two real roots. Parabola  crosses x-axis two times.	Example: $\eqalign{x^2-2x+1=0\\\text{Plug into discriminant}\\(-2)^2-4(1)(1)\\4-4\\0}$ One real root. Parabola crosses x-axes one time (vertex on x-axis)	Example: $\eqalign{x^2-2x+9=0\\\text{Plug into discriminant}\\(-2)^2-4(9)(1)\\4-36\\-32}$ Two imaginary roots. Prabola does not cross x-axis.

So, overall, knowing the discriminant of a quadratic can help you see how many solutions the parabola will have, which tells you how many times the parabola crosses the x-axis.