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Quadratic Functions

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 by factoring.
Solution: The integer factors of -2 are 1 and 2 with one of them being negative. So we try

(Not correct)

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
number x for which That is has no zeros.
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

two zeros
one zero
no zeros

Definition: The expression   is called the discriminate of the function
Example 5: Calculate the discriminate of and plot the function .
Solution:

Since the discriminate is positive we know that has two zeros
and 4.37
Example 6: Calculate the discriminate of  and plot the function.
Solution:



Since the discriminate is zero, there is only one zero, and it equals
Example 7: Calculate the discriminate of   and plot the function.

Solution:

Since the discriminate is negative, this quadratic function has no zeros.
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