| Zeros of Quadratic Functions
In this page we define what is meant by a zero of a quadratic function and how to find all of them.
By a zero of a quadratic function we mean a number x 0 such that
|There are two ways to find the zeros of a
quadratic function. The first, and easiest, is to factor the quadratic
expression if you can . The second, and this always works, is to use the quadratic formula. Recall, if the
expression equals zero, then
|Example 1: Find the zeros of
Solution: The integer factors of -2 are 1 and 2 with one of them being negative. So we try
However, when we multiply the two factors together to see if we’ve got it correct, we compute
which is not what we want. The coefficient of x is off by a minus sign. So we try
and this is correct. The zeros of are the solutions of the following equation
The only way a product can equal zero is for one of the factors to equal zero. Thus, we have
From which we conclude that
Notice that there are exactly two zeros.
|Example 2: What are the zeros of
Solution: Thus the zeros of this quadratic function are
|Example 3: Find the zeros of
by using the quadratic formula.
Solution: We are looking to find those x for which The solutions of this equation are
given by the quadratic formula
|We now discuss why some quadratic functions have
no zeros. If we graph the quadratic function
we see that it never crosses the x-axis. This means that f x never equals zero, or this function
has no zeros.
It is instructive to use the quadratic formula to find the zeros of .
Since we cannot take the square root of a negative
number and get a real number, we see that there is no real
|Example 4: Does
have any zeros?
Solution: Using the quadratic formula to solve the equation we have :
Since negative numbers do not have real square roots, this quadratic function has no zeros.
|In the table below we summarize the possibilities
of zeros for an arbitrary quadratic function
Definition: The expression
is called the
discriminate of the function
|Example 5: Calculate the discriminate of
and plot the function .
Since the discriminate is positive we know that
has two zeros
|Example 6: Calculate the discriminate of
and plot the function.
Since the discriminate is zero, there is only one zero, and it equals
|Example 7: Calculate the discriminate of
and plot the function.
Since the discriminate is negative, this quadratic function has no zeros.