Lesson 7 Taking the Scenic Root Develop Understanding

Learning Focus

Solve quadratic equations graphically and algebraically.

Make connections between solving quadratic equations and graphing quadratic functions.

How can we solve quadratic equations that can’t be factored easily?

How can we use graphs to solve quadratic equations?

Open Up the Math: Launch, Explore, Discuss

At Optima’s Quilts, the employees are always thinking about the relationship between the area of a square or rectangle and the lengths of the sides. When new employees are being trained, they work only with squares, using the relationship $A (x) = x^{2}$ . Joon and Linnea are just beginning, and the trainer asks them to create a table that they can refer to whenever they need to find the area of a square when they know the length of a side.

1.

Help Joon and Linnea by making the table the trainer has asked for.

2.

The trainer is pleased with their work and asks, “How can you use the chart to find the length of a side if you know the area?” How would you answer this question?

3.

When Joon and Linnea seem to be comfortable using the table to find both sides and area, the trainer asks, “What is the length of each side if the area of the square is $12$ square inches?” How would you answer this question?

4.

After thinking for a few minutes, Joon decided to write the equation $x^{2} = 12$ . The equation helped him to see that the solution needed to be a number that could be squared to get $12$ for an answer. Joon said, “Hmm, it’s not a number on my table, but it must be between numbers on my table.” Explain how Joon could find the number he is looking for.

5.

Linnea said, “Making a table from an equation makes me think about my favorite representation, a graph. I’m going to use a graph to solve this problem.” The graph below shows what Linnea found when she graphed the function. (Does it look familiar?) How could Linnea use the graph to find the length of the side?

6.

Both Joon and Linnea find that the length of the side is about $3.46 inches$ . Explain how this number relates to $\sqrt{12}$ .

7.

When Linnea looks at the graph, she notices that $- 3.46$ would also be a solution. Explain why there are two numbers that can be squared to get $12$ .

8.

Find both solutions to the equations:

a.

$x^{2} = 15$

b.

$x^{2} = 27$

c.

$x^{2} = 144$

Now that Joon and Linnea have a little experience, the trainer brings up the idea that sometimes the sides of squares get extended so that the side of a square might be $(x + 1)$ , making the area of the square ${(x + 1)}^{2}$ . The trainer wants to know what values of $x$ makes the area of the square $9$ .

Joon and Linnea both go to work to write an equation and draw a graph, using their preferred methods. Joon writes:

${(x + 1)}^{2} = 9$

$\sqrt{{(x + 1)}^{2}} = \pm \sqrt{9}$

$x + 1 = \pm 3$

$x = - 1 \pm 3$

$x = 2$ or $x = - 4$

9.

Use the space to the right of Joon’s work to explain each step of Joon’s algebraic method for finding the possible values of $x$ .

Step 1:	${(x + 1)}^{2} = 9$
Step 2:	$\sqrt{{(x + 1)}^{2}} = \pm \sqrt{9}$
Step 3:	$x + 1 = \pm 3$
Step 4:	$x = - 1 \pm 3$
Step 5:	$x = 2$ or $x = - 4$

10.

Linnea began by finding the graph of $A (x) = {(x + 1)}^{2}$ , which is shown below. She then drew a line where $A (x) = 9$ . Explain how to use Linnea’s graph to find the two solutions.

11.

When Linnea and Joon compared their two methods, they noticed something interesting. Linnea said, “I see that the line of symmetry for this parabola is $x = - 1$ . It looks like the solutions are both $3$ units away from the line of symmetry.” Joon said, “That matches up with the way I solved the equation.” What are the similarities between the methods that Linnea and Joon could be noticing?

12.

Joon was excited and wanted to see if this was always the case. He made up this example:

“Let’s say that the side of a square was reduced by $3$ , making the length of the side $(x - 3)$ , and the area of the square is ${(x - 3)}^{2}$ . If we know that the area of the square is $25$ , then what are the possible values for $x$ ?” Answer Joon’s question using both algebraic and graphical methods.

13.

Compare the two methods. Do you see a similar pattern emerging?

14.

Linnea said that she wants to try it with an area that is a little harder to think about. She comes up with this example: Let’s say that the side of a square was reduced by $2$ , making the length of the side $(x - 2)$ , and the area of the square is ${(x - 2)}^{2}$ . If we know that the area of the square is $20$ , then what are the possible values for $x$ ?” Answer Linnea’s question using both algebraic and graphical methods.

15.

When Joon worked this problem algebraically, he found three different ways to write his answers:

$x = 2 \pm \sqrt{20}$

$x = 2 \pm 2 \sqrt{5}$

$x = 4.24$ and $x = - .24$

a.

Are these answers equivalent? Justify your answer.

b.

How do each of these answers relate to the graphical method of finding values of $x$ ?

16.

Solve each of these equations using both algebraic and graphical methods to check your work. Write your answers two ways, with a square root that does not contain perfect square factors and with decimal approximations.

a.

${(x + 4)}^{2} = 28$

b.

${(x - 1)}^{2} = 49$

c.

${(x + 5)}^{2} = 27$

Ready for More?

Find two methods for solving this equation graphically:

$3 x^{2} + 3 x - 5 = - x + 7$

Takeaways

Solving quadratic equations using inverse operations:

Example	Procedure
${(x + 2)}^{2} = 16$	Given.

Solving quadratic equations by graphing:

Example	Procedure
${(x + 2)}^{2} = 16$	Given.

Lesson Summary

In this lesson, we learned methods for solving quadratic equations. Some quadratic equations can be solved using inverse operations and taking the square root of both sides of the equations. Some quadratic equations can be solved by factoring and using the Zero Product Property. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

Retrieval

Rewrite the equations for the area of a rectangle to reveal the side lengths of the rectangle.

1.

$A (x) = x^{2} - 49$

2.

$A (x) = x^{2} - 16$

Write the equation for the quadratic function represented in the graph in factored form.