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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:


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 $D$.




So, the discriminant of this equation is 100.  

What can we learn from that?

D>0 (positive)D=0D<0 (negative)
Parabola has 2 unequal real rootsParabola has 1 real rootParabola has two imaginary roots
Parabola with positive discriminant

Parabola with 0 discriminant

Parabola with negative discriminant


$\eqalign{x^2+6x-2=0\\\text{Plug into discriminant}\\6^2-4(1)(-2)\\36+8\\44}$

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.  

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

Common Core Grade Level/Subject

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