*ax*^{2} + *bx* + *c* = 0

*a, b, c *are constants (generally integers)

Synonyms: Solutions or Zeros

- Can have 0, 1, or 2 real roots

Consider the graph of quadratic equations. The quadratic equation looks like
*ax*^{2} + *bx* + *c* = 0,
but if we take the quadratic *expression* on the left and set it equal to *y*,
we will have a function:

*y* = *ax*^{2} + *bx* + *c*

When we graph *y* vs. *x*, we find that we
get a curve called a *parabola*. The specific values of *a*,
*b*, and *c* control where the curve is relative to the origin (left,
right, up, or down), and how rapidly it spreads out. Also, if *a* is
negative then the parabola will be upside-down. What does this have to do with
finding the solutions to our original quadratic equation? Well, whenever *y* = 0
then the equation *y* = *ax*^{2} + *bx* + *c*
is the same as our original equation.

Graphically, *y* is zero whenever the curve crosses the *x*-axis.
Thus, the solutions to the original quadratic equation (*ax*^{2} + *bx* + *c*
= 0) are the values of *x* where the function (*y* = *ax*^{2} + *bx* + *c*)
crosses the *x*-axis. From the figures below, you can see that it can
cross the *x*-axis once, twice, or not at all.

Actually, if you have a graphing calculator
this technique can be used to find solutions to
- Move all the terms to one side, so that it is equal to zero
- Set
the resulting expression equal to
*y*(in place of zero) - Enter the function into your calculator and graph it
- Look
for places where the graph crosses the
*x*-axis
Your graphing calculator most likely has a function that will
automatically find these intercepts and give you the |