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

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Julio is solving three equations by graphing.

$(x - 5) (x - 3) = 0$

$(x - 5) (x - 3) = - 1$

$(x - 5) (x - 3) = - 4$

To solve the first equation, $(x - 5) (x - 3) = 0$ , he graphed $y = (x - 5) (x - 3)$ and then looked for the $x$ -intercepts of the graph.

1. Explain why the $x$ -intercepts can be used to solve $(x - 5) (x - 3) = 0$ .

Compare your answer:

The solutions to $(x - 5) (x - 3) = 0$ are $x$ -values that make the expression have a value of 0. At the $x$ -intercepts, the $y$ -value of the expression $(x - 5) (x - 3)$ is 0, so the $x$ -intercepts give the solutions.

2. Graph $y = (x - 5) (x - 3)$ . What are the solutions?

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Compare your answer:

The parabola opens up and has a vertex at the point (4, negative 1) the x-intercepts are (3, 0) and (5, 0).

The solutions are $x = 5$ and $x = 3$ .

To solve the second equation, Julio rewrote it as $(x - 5) (x - 3) + 1 = 0$ . He then graphed $y = (x - 5) (x - 3) + 1$ .

3. Graph $y = (x - 5) (x - 3) + 1$ . Then, use the graph to solve the equation. Be prepared to explain how you used the graph for solving.

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Compare your answer:

The parabola opens up and has a vertex at the point (4, 0) the x-intercepts is (4, 0).

The solution is 4 for $(x - 5) (x - 3) + 1 = 0$ . The solution is the $x$ -coordinate of the $x$ -intercept.

Solve the third equation using Julio's strategy of graphing the equations.

4. What are the solutions to $y = (x - 5) (x - 3) + 4$ ?

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Compare your answer:

The parabola opens up and has a vertex at the point (4, 3)the x-intercepts is (4, 3).

There are no solutions for $(x - 5) (x - 3) + 4 = 0$ . The graph has no $x$ -intercept.

Think about the graphing strategy used to solve the equations and the solutions you found. View the Desmos graphs of all three equations.

A graph with three upward-opening parabolas in red, blue, and green on a grid. All have similar shapes and are centered around x equals 4, but each reaches its minimum at different y-values.

5. Why might it be helpful to rearrange each equation to equal 0 on one side and then graph the expression on the non-zero side?

Compare your answer:

When a quadratic expression is equal to 0 and we want to find the unknown values, we can think of it as finding the zeros of a quadratic function, or finding the $x$ -intercepts of the graph of that function.

6. How many solutions does each of the three equations have?

Enter the number of solutions for each equation.

Compare your answer:

There are two solutions for the first equation, one solution for the second equation, and no solution for the last equation.

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Extending Your Thinking

1.

Graph $y = (x - 3) (x - 5)$ and $y = - 1$ on the same coordinate plane.

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How do the equations $y = (x - 3) (x - 5)$ and $y = - 1$ help solve the equation $(x - 3) (x - 5) = - 1$ ?

Compare your answer:

The solution to $(x - 3) (x - 5) = - 1$ is the $x$ -value of the intersection point of the two graphs of $y = (x - 3) (x - 5)$ and $y = - 1$ . This solution is 4.

Graph of a parabola that opens up with a vertex at (4, negative 1) and x-intercepts of 3 and 5a horizontal dotted line is graphed aty equals negative 1.

2.

Graph $y = (x - 3) (x - 5)$ and $y = 0$ on the same coordinate plane.

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How do the equations $y = (x - 3) (x - 5)$ and $y = 0$ help solve the equation $(x - 3) (x - 5) = 0$ ?

Compare your answer:

The solutions to $(x - 3) (x - 5) = 0$ are the $x$ -values of the intersection points of the two graphs of $y = (x - 3) (x - 5)$ and $y = 0$ . These solutions are 3 and 5.

Graph of a parabola that opens up with a vertex at (4, negative 1) and x-intercepts of 3 and 5a horizontal dotted line is graphed at y equals 0.

3.

Graph $y = (x - 3) (x - 5)$ and $y = 3$ on the same coordinate plane.

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How do the equations $y = (x - 3) (x - 5)$ and $y = 3$ help solve the equation $(x - 3) (x - 5) = 3$ ?

Compare your answer:

The solutions to $(x - 3) (x - 5) = 3$ are the $x$ -values of the intersection points of the two graphs of $y = (x - 3) (x - 5)$ and $y = 3$ . These solutions are 2 and 6.

Graph of a parabola that opens up with a vertex at (4, negative 1) and x-intercepts of 3 and 5 a horizontal dotted line is graphed at y equals 3.

4.

4. Use the graphs you created in questions 1 – 3 to help you find a few other equations of the form $(x - 3) (x - 5) = z$ that have whole-number solutions.

Compare your answer:

There are many different answers. For example:

$(x - 3) (x - 5) = 8$ , $(x - 3) (x - 5) = 15$

$(x - 3) (x - 5) = 24$

5.

5. Some of the values of $z$ that you might have found are –1, 0, 3, 8, 15, 24, etc. Find a pattern in the values of $z$ that give whole-number solutions.

Compare your answer:

Starting at $x = - 1$ , add 1, then 3, then 5, and so on, to find the next $x$ -value that yields whole-number solutions.

Because the expressions $x - 3$ and $x - 5$ represent numbers whose difference is 2, the values of $z$ have the form $- 1 \cdot 1$ , $0 \cdot 2$ , $1 \cdot 3$ , $2 \cdot 4$ , and so on.

6.

6. Without solving, determine if $(x - 5) (x - 3) = 120$ and $(x - 5) (x - 3) = 399$ have whole-number solutions. Be prepared to show your reasoning.

Compare your answer:

Both have whole-number solutions. The values of $x$ that produce whole-number solutions are all 1 less than a square number.

Self Check

Use graphing technology to determine the number of solutions for the equation

(x+3)(x−6)=−22

. Then find any solutions.

The equation has no solutions.
There are two solutions. $x=-3$ or $x=6$
There are two solutions. $x=-6$ or $x=3$
There is one solution. $x=5.5$

Additional Resources

Interpreting Graphs to Solve Quadratic Equations

Let's analyze and interpret the graphs of different quadratic equations.

Example 1

First, let's look at the equation $(x - 4) (x - 7) = 0$ .

We will graph the equation $y = (x - 4) (x - 7)$ .

Graph of a parabola that opens up and passes through the points (4, 0) and (7, 0).

This graph has two $x$ -intercepts.

The solutions to $(x - 4) (x - 7) = 0$ are $x$ -values that make the expression have a value of 0.

At the $x$ -intercepts, the $y$ -value of the expression $(x - 4) (x - 7)$ is 0, so the $x$ -intercepts give the solutions.

The $x$ -intercepts and solutions to the equation are 4 and 7.

Note that the values of 4 and 7 could be referred to as solutions of the equation, $x$ -intercepts of the graph, roots of the equation, or zeros of the equation.

Example 2

Let's look at a different example.

$(x - 4) (x - 7) = - 2.25$

First, we rewrite the equation to set it equal to 0.

$(x - 4) (x - 7) + 2.25 = 0$

We will graph the equation $y = (x - 4) (x - 7) + 2.25$ .

Graph of a parabola that opens up with a vertex at (5.5, 0).

This graph has one $x$ -intercept. It is an example of a double root at 5.5. A double root is a root that appears twice in the solution of an algebraic equation.

The solution is the $x$ -coordinate of the $x$ -intercept.

So, the solution to $(x - 4) (x - 7) = - 2.25$ is 5.5.

Example 3

Let's look at one last example.

$(x - 4) (x - 7) = - 5$

First, we rewrite the equation to set it equal to 0.

$(x - 4) (x - 7) + 5 = 0$

We will graph the equation $y = (x - 4) (x - 7) + 5$ .

Graph of a parabola that opens up the graph does not touch or cross the x-axis the graph passes through the points (4, 5) and (7, 5).

The graph has no $x$ -intercepts.

So, there are no real solutions for $(x - 4) (x - 7) + 5 = 0$ .

Remember, when a quadratic expression is equal to 0 and we want to find the unknown values, we can think of it as finding the zeros of a quadratic function, or finding the $x$ -intercepts of the graph of that function.

Try it

Try It: Interpreting Graphs to Solve Quadratic Equations

Determine the number of solutions for the equation $(x - 4) (x - 9) = - 6.25$ . Then find any solutions.

Use the Desmos graphing tool or technology outside the course.

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Compare your answer:

Here is how to analyze the graph of the equation to find any solutions:

Step 1 - Rewrite the equation to set it equal to 0.

$(x - 4) (x - 9) + 6.25 = 0$

Step 2 - Graph the equation $y = (x - 4) (x - 9) + 6.25$ .

Graph of a parabola that opens up with a vertex at the point (6.5, 0).

Step 3 - Identify the $x$ -intercepts from the graph.

There is one $x$ -intercept at $(6.5, 0)$

This means there is one solution.

The solution is 6.5.

8.5.2 Interpreting Graphs to Solve Quadratic Equations

Activity

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Extending Your Thinking

Additional Resources

Interpreting Graphs to Solve Quadratic Equations

Try It: Interpreting Graphs to Solve Quadratic Equations