Lesson 3 Finding Your Roots Solidify Understanding

Jump Start

Solve the following equation. Show all of your steps.

$x^{2} + 9 = 5 (x + 1)$

Learning Focus

Solve equations and systems of equations that involve square root expressions.

How do I adapt my strategies for solving equations and systems of equations when the equations include square root expressions?

Why do my equation solving strategies for square root equations sometimes introduce solutions that do not satisfy the original equation?

Open Up the Math: Launch, Explore, Discuss

Trevor has been reviewing some of the adjustments Taska has made to the swings for the amusement park on Jupiter. Taska has recorded the the final period of the swing after the adjustments have been made, but Trevor realizes that he needs to know the length of the original swing before the adjustments were made.

For example, Tehani had written a set of instructions for Taska that looked like this: $T = 0.6 \sqrt{x + 5}$ . After making the adjustments, Taska measured the period of the swing and recorded the results like this: $3.6 = 0.6 \sqrt{x + 5}$ . Now Trevor has to find the length of the original swing.

Trevor begins by interpreting Tehani’s instructions, hoping this will help him find a strategy for solving for $x$ , the length of the original swing:

Add $5 feet$ to the length of the original swing	$x + 5$
Take the square root of the new length	$\sqrt{x + 5}$
Multiply by the constant $k = 0.6$	$0.6 \sqrt{x + 5}$
The product is the new period of $3.6 seconds$	$3.6 = 0.6 \sqrt{x + 5}$

1.

Using these instructions, find the length of the original swing before Taska made adjustments. Show your steps and explain how you are using Tehani’s instructions to solve for $x$ .

Here are some additional swing adjustment equations. Write an explanation to convince Trevor that you have found the correct original length of the associated swing. Remember, the fate of the engineering competition rests in your hands!

2.

$5.0 = 0.6 \sqrt{x} + 2$

3.

$2.4 = 0.6 \sqrt{2 x}$

4.

$5.4 = 0.6 \sqrt{2 x} + 3$

Trevor’s friend Mateo is on the engineering team in charge of planning the escape routes for the Tarzan: Escape from the Jungle swing ride. They want the routes to intersect at various points during the ride in order to increase the thrill of the riders as they encounter potential collisions with other riders. Of course, these impending collisions will be carefully timed in order to avoid actual danger. Mateo has come to Trevor to discuss the mathematics of modeling some of the escape routes, since he is less familiar with the square root functions that are being used in the mathematical description of some of the routes.

Mateo needs to find the intersection points of the two escape routes modeled by the equations $y = \sqrt{2 x - 4}$ and $y = x - 6$ . These routes are being modeled on a coordinate grid with Tarzan’s treehouse located at the origin of the grid.

Here is the discussion between Trevor and Mateo:

Mateo: I treated the two equations as the system of equations ${\begin{matrix} y = \sqrt{2 x - 4} \\ y = x - 6 \end{matrix}$ , since I wanted to find the points of intersection of the two graphs. I first tried to solve the system by graphing the two equations and this is what I got: (see Mateo’s graph).

5.

Based on Mateo’s graph, what is the solution to this system of equations?

Trevor: That square root graph looks strange. I wonder why it starts at $(2, 0)$ instead of $(0, 0)$ ?

6.

How would you answer Trevor’s question?

Mateo: I wanted to verify the solution using algebra, since I know that reading a solution from a graph can be somewhat inaccurate. I started by setting the expressions equal to each other, like this: $\sqrt{2 x - 4} = x - 6$ . Then I started to solve this equation by squaring both sides. I ended up with a quadratic equation, which I knew how to solve, but the result confused me.

7.

Solve the equation $\sqrt{2 x - 4} = x - 6$ using Mateo’s suggested strategy.

Trevor: I wonder why you got two different solutions for $x$ , when the graph clearly only shows one solution to the system?

8.

How would you explain the issue that has confused Trevor and Mateo? Was their algebra incorrect, or has something else created this extra solution? (Note: The extra solution that was created during the solution process is called an extraneous solution.)

9.

Mateo knew that he had an extraneous solution by examining the graph of the system. If he didn’t want to create an accurate graph of the system, how else might he check his algebraic work for extraneous solutions?

Pause and Reflect

10.

Help Mateo solve this system: ${\begin{matrix} y = \sqrt{3 x - 2} + 1 \\ y = x - 3 \end{matrix}$

Make sure you check for extraneous solutions.

11.

Help solve this problem for Trevor. What would the solution to this equation mean in terms of lengths of swings located on Earth and Jupiter?

$1.2 \sqrt{L} = 0.6 \sqrt{2 L + 30}$

Ready for More?

Is it possible to create an equation containing both a linear and a square root expression for which all potential solutions that result from an appropriate equation-solving strategy are extraneous? Try to find such an equation or explain why it is not possible.

Takeaways

When solving an equation that involves a square root expression:

Vocabulary

extraneous solution
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to solve equations that include square root expressions. The process we used often required us to recall how to solve quadratic equations. Sometimes the process introduced extraneous solutions, so it is important to check the solutions to make sure they satisfy the original equation.

Retrieval

1.

Two functions are represented in different ways. Identify the type of function as linear, quadratic, or square root. Give the domain and range of the specific function in the problem. Write the equation of the parent function. Then give the domain and range of the parent function. Describe the rate of change of this type of function.

Function A:

$x$	$y$
$- 2$	$- 2$
$- 1$	$- 5$
$0$	$- 6$
$1$	$- 5$
$2$	$- 2$
$3$	$3$

Function B:

Function A:

type of function:

domain:

range:

description of the equation or form:

domain:

range:

describe the rate of change:

Function B:

type of function:

domain:

range:

description of the equation or form:

domain:

range:

describe the rate of change:

2.

Solve: $- \frac{2}{5} x - 7 > 9$

3.

Solve: $54 + x < - 17 x$