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 xaxis. 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 . Solution:
Since the discriminate is positive we know that
has two zeros


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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