Mastery Check Solutions

Show What You Know

The ratio of the length to width of a certain rectangle is: $\frac{\sqrt{2 x} + 5}{\sqrt{2 x} + 1}$

Using the given side lengths, determine the perimeter of the rectangle.

$\begin{array}{rcl} P & = & 2 (l + w) \\ P & = & 2 (\sqrt{2 x} + 5 + \sqrt{2 x} + 1) \\ P & = & 2 (2 \sqrt{2 x} + 6) \\ P & = & 4 \sqrt{2 x} + 12 \end{array}$

Note

Read the first sentence carefully. This gives the length and width of the rectangle in a different form, but one you should be familiar with.

The area of the same rectangle is 5x + 5 square meters. Solve for x.

$\begin{array}{rcl} A & = & l w \\ 5 x + 5 & = & (\sqrt{2 x} + 5) (\sqrt{2 x} + 1) \\ 5 x + 5 & = & 2 x + \sqrt{2 x} + 5 \sqrt{2 x} + 5 \\ 5 x + 5 & = & 2 x + 6 \sqrt{2 x} + 5 \\ 3 x & = & 6 \sqrt{2 x} \\ \frac{x}{2} & = & \sqrt{2 x} \end{array}$

$\begin{array}{rcl} {(\frac{x}{2})}^{2} & = & {(\sqrt{2 x})}^{2} \\ \frac{x^{2}}{4} & = & 2 x \\ x^{2} & = & 8 x \\ x^{2} - 8 x & = & 0 \\ x (x - 8) & = & 0 \\ x & = & 0, 8 \end{array}$

Note

Q: What property do you need to use to multiply two binomials together?

A: Distributive property
Be sure to square the numerator and denominator of the fraction as you solve for x.

Determine the value of the length and width of the rectangle.

When x = 0

$l = \sqrt{2 (0)} + 5 l = 5 w = \sqrt{2 (0)} + 1 w = 1$

When x = 8

$l = \sqrt{2 (8)} + 5 l = \sqrt{16} + 5 l = 9$
$w = \sqrt{2 (8)} + 1 w = \sqrt{16} + 1 w = 5$

Note

Q: Why is it possible for zero to be a value of x in this problem?

A: Because when you substitute zero into the expressions for the side lengths, the side is greater than zero.

Q: What would the area be when x = 0?

A: The answer will be 5 square meters.

Q: What would the perimeter be when $x = 8$ ?

A: 28 meters

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

- Solve a radical equation.
- Solve a radical equation written with rational (fractional) exponents.
- Determine if there are any extraneous solutions to a radical equation.