The solutions to *y* = *f* (*x*) when *y* = 0 are called the roots of a
function (*f* (*x*) is any function). These
are the points at which the graph of an equation crosses the *x*-axis.

###
Roots of Quadratic Functions

We have already learned to solve for *x* in *ax*^{2} + *bx* + *c* = 0 by
factoring *ax*^{2} + *bx* + *c* and
using the zero product property.
Since the roots of a function are the points at which *y* = 0, we can
find the roots of *y* = *ax*^{2} + *bx* + *c* = 0 by factoring *ax*^{2} + *bx* + *c* = 0 and solving for *x*. We can also find the roots of *y* = *ax*^{2} + *bx* + *c* = 0 using the quadratic formula,
and we can find the number of roots using the
discriminant.

If a quadratic function has 2 roots--i.e., if it can be factored into
2 distinct binomials or if *b*^{2} -4*ac* > 0--then it crosses the *x*-
axis twice. Either the vertex
is below the *x*-axis and the leading
coefficient is positive, or the
vertex is above the *x*-axis and the leading coefficient is negative.

If a quadratic function has 1 root (a "double root")--i.e. if it can
be factored as the square of a single binomial or if *b*^{2} - 4*ac* = 0--then
it crosses the *x*-axis once. The vertex lies on the *x*-axis, and
the leading coefficient can be positive or negative.

If a quadratic function has no roots--i.e. if it cannot be factored or
if *b*^{2} -4*ac* < 0--then it does not cross the *x*-axis. Either the
vertex is above the *x*-axis and the leading coefficient is positive, or
the vertex is below the *x*-axis and the leading coefficient is
negative. The quadratic equation is said to have 2
imaginary roots.

###
Roots of Other Polynomial Functions

We can find the roots of other polynomial functions by setting *y* = 0
and factoring. For example, *y* = *x*^{3} -27 = (*x* - 3)(*x*^{2} +3*x*^{2} + 9)
has one root (*x* = 3), because there is one value of *x* for which *x* - 3 = 0 and no values of *x* for which *x*^{2} + 3*x* + 9 = 0 (the
discriminant is negative). *y* = 45*x*^{3} +18*x*^{2} - 5*x* - 2 = (3*x* + 1)(3*x* - 1)(5*x* + 2) has three roots (*x* = - ,, - ).