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

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

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

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:

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.

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.

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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.

Solve $x^{2} - 2 x + 1 = 0$ by using the factored form and zero product property. What solutions do you get?

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.

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

Use the Desmos graphing tool or technology outside the course.

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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.

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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

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

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 $(- 2, 0)$ .

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 $(- 10, 0)$ .