# 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 |

$\eqalign{x^2+6x-2=0\\\text{Plug into discriminant}\\6^2-4(1)(-2)\\36+8\\44}$ Two real roots. Parabola crosses x-axis two times. |
$\eqalign{x^2-2x+1=0\\\text{Plug into discriminant}\\(-2)^2-4(1)(1)\\4-4\\0}$ One real root. Parabola crosses x-axes one time (vertex on x-axis) |
$\eqalign{x^2-2x+9=0\\\text{Plug into discriminant}\\(-2)^2-4(9)(1)\\4-36\\-32}$ 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.

#### Practice Problems:

## Discriminant

Using the discriminant, find out if the following equations have 1, 2, or no real solutions.

- $x^2-7x+9=0$
- $2x^2+4x-10=0$
- $4x^2-x=-20$
- $x^2=4x-8$
- $x^2+6=-3x$
- $2x^2-6x+6=0$
- $-6x+2=2x^2$
- $5x^2-2x+1=0$
- $6x^2+6x+6=0$
- $3x^2-12x=-12$

#### Answer Key: