Activity
In previous lessons, we saw that a quadratic equation can have zero, one, or two solutions.
Sketch graphs that represent three quadratic functions: one that has no zeros, one with one zero, and one with two zeros.
Compare your answers:
No zeros:
One zero:
Two zeros:
Graph the function defined by . Examine the -intercepts of the graph.
Use the Desmos graphing tool or technology outside the course.
Explain what the -intercepts reveal about the function.
Compare your answer:
The graph only intersects the -axis at one point: . The function only has one zero.
Solve by using the factored form and zero product property. What solutions do you get?
Enter the solutions.
Compare your answer:
Rewrite the standard form to . The solutions are and . Because the two values are the same, we say that this equation has one solution and that is a double root for the equation.
Try to find another quadratic function that you think will only have one zero.
Use the Desmos graphing tool or technology outside the course.
Graph it to check your prediction, then enter your final answer.
Compare your answer:
For example: . When graphed, this function only has one zero, at .
Video: Quadratic Equations with Only One Solution
Watch the following video to learn more about writing an equation to represent a quadratic function with only one solution.
Self Check
Additional Resources
Writing an Equation to Represent a Quadratic Function with Only One Solution
Identifying the Number of Zeros
How many zeros does the graph of the function have? How can you find out without graphing the function?
Let's find the factored form of the function and solve it.
Step 1 - Set the function equal to 0.
Step 2 - Find the factored form.
Step 3 - Set each factor equal to 0 and solve.
and
Since there are two solutions, we know that the function will have zeros at and .
In other words, the -intercepts of the function are and .
For any function, the number of solutions is equal to the number of zeros when graphed.
Writing an Equation to Represent a Quadratic Function with One Solution
We can also find a quadratic function if we know the zeros. For example, if we want to write a quadratic function that has one zero or one solution, we can start with a factored form of the function representing one zero or solution.
Any factored form of a function with a factor that repeats itself, such as , has only one solution or one zero.
We can graph the function to check our prediction.
We know the solution is . Let's rewrite the function in standard form.
Factored form:
Multiply to find standard form:
Now, we graph the function .
The graph of the function has one zero at . The -intercept of the function is . The equation is said to have a double root at this point.
So, our prediction was correct.
Try it
Try It: Writing an Equation to Represent a Quadratic Function with Only One Solution
Write a quadratic function in standard form whose graph has two zeros.
Here is how to write the quadratic functions in standard form:
The quadratic function has two solutions.
This means the factors in factored form are not equal. We could use: .
Multiply to find standard form.
The graph of has two zeros. The -intercepts will be and .
Write a quadratic function in standard form whose graph has one zero.
Here is how to write the quadratic functions in standard form:
The quadratic function has one solution.
That means the factors in factored form are equal. We could use:
Multiply to find standard form.
The graph of has one zero. The -intercept will be .