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 ![]() ![]() |
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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 ![]()
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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 ![]() ![]() 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. |
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Example 2: What are the zeros of
![]() Solution: ![]() ![]() |
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Example 3: Find the zeros of
![]() Solution: We are looking to find those x for which ![]() given by the quadratic formula ![]() |
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We now discuss why some quadratic functions have
no zeros. If we graph the quadratic 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 |
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Example 4: Does
![]() Solution: Using the quadratic formula to solve the equation ![]() Since negative numbers do not have real square roots, this quadratic function has no zeros. |
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In the table below we summarize the possibilities
of zeros for an arbitrary quadratic function
Definition: The expression
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Example 5: Calculate the discriminate of
![]() Solution:
Since the discriminate is positive we know that
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Example 6: Calculate the discriminate of
![]() Solution: ![]() Since the discriminate is zero, there is only one zero, and it equals ![]() |
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Example 7: Calculate the discriminate of
![]() Solution: ![]() Since the discriminate is negative, this quadratic function has no zeros. |
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