Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

In previous lessons, we saw that a quadratic equation can have zero, one, or two solutions.

1.

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:

A graph of an upward-opening parabola on the x and y axes, showing part of the curve above the x-axis with the vertex at the lowest point.

One zero:

A graph of a parabola opening upwards with its vertex marked by a black dot, centered at the origin on the x-y coordinate plane.

Two zeros:

A graph of a parabola opening upward, intersecting the x-axis at two points. The x and y axes are labeled, and the intersection points are marked with solid black dots.

2.

Graph the function defined by $f (x) = x^{2} - 2 x + 1$ . Examine the $x$ -intercepts of the graph.

Use the Desmos graphing tool or technology outside the course.

Access multimedia content

Explain what the $x$ -intercepts reveal about the function.

Compare your answer:

The graph only intersects the $x$ -axis at one point: $(1, 0)$ . The function only has one zero.

3.

Solve $x^{2} - 2 x + 1 = 0$ 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 $(x - 1) (x - 1) = 0$ . The solutions are $x = 1$ and $x = 1$ . Because the two values are the same, we say that this equation has one solution and that $x = 1$ is a double root for the equation.

4.

Try to find another quadratic function that you think will only have one zero.

Use the Desmos graphing tool or technology outside the course.

Access multimedia content

Graph it to check your prediction, then enter your final answer.

Compare your answer:

For example: $f (x) = (x + 4) (x + 4)$ . When graphed, this function only has one zero, at $x = - 4$ .

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.

Access multimedia content

Self Check

Without graphing, identify the function whose graph has only one zero.

$f(x) = x^2+ 2x - 15$
$f(x) = x^2-14x + 49$
$f(x) = x^2- 1$
$f(x) = x^2- 7x - 18$

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 $f (x) = x^{2} + 5 x - 14$ 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.

$x^{2} + 5 x - 14 = 0$

Step 2 - Find the factored form.

$(x + 7) (x - 2) = 0$

Step 3 - Set each factor equal to 0 and solve.

$x = - 7$ and $x = 2$

Since there are two solutions, we know that the function $f (x) = x^{2} + 5 x - 14$ will have zeros at $x = - 7$ and $x = 2$ .

In other words, the $x$ -intercepts of the function are $(- 7, 0)$ and $(2, 0)$ .

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 $(x - 3) (x - 3)$ , has only one solution or one zero.

We can graph the function $f (x) = (x - 3) (x - 3)$ to check our prediction.

We know the solution is $x = 3$ . Let's rewrite the function in standard form.

Factored form:

$f (x) = (x - 3) (x - 3)$

Multiply to find standard form:

$f (x) = x^{2} - 6 x + 9$

Now, we graph the function $f (x) = x^{2} - 6 x + 9$ .

Graph of a parabola that opens up with a vertex at the point (3, 0).

The graph of the function has one zero at $x = 3$ . The $x$ -intercept of the function is $(3, 0)$ . 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

1.

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: $f (x) = (x + 5) (x + 2)$ .

Multiply to find standard form. $f (x) = x^{2} + 7 x + 10$

The graph of $f (x) = x^{2} + 7 x + 10$ has two zeros. The $x$ -intercepts will be $(- 5, 0)$ and $(- 2, 0)$ .

2.

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: $f (x) = (x + 10) (x + 10)$

Multiply to find standard form. $f (x) = x^{2} + 20 x + 100$

The graph of $f (x) = x^{2} + 20 x + 100$ has one zero. The $x$ -intercept will be $(- 10, 0)$ .

8.9.3 Writing an Equation to Represent a Quadratic Function with Only One Solution

Activity

Video: Quadratic Equations with Only One Solution

Additional Resources

Writing an Equation to Represent a Quadratic Function with Only One Solution

Try It: Writing an Equation to Represent a Quadratic Function with Only One Solution