# Lesson 21

Sums and Products of Rational and Irrational Numbers

• Let’s make convincing arguments about why the sums and products of rational and irrational numbers always produce certain kinds of numbers.

### Problem 1

Match each expression to an equivalent expression.

(From Unit 7, Lesson 15.)

### Problem 2

Consider the statement: "An irrational number multiplied by an irrational number always makes an irrational product."

Select all the examples that show that this statement is false.

A:

$$\sqrt4\boldcdot\sqrt5$$

B:

$$\sqrt4\boldcdot\sqrt4$$

C:

$$\sqrt7\boldcdot\sqrt7$$

D:

$$\frac{1}{\sqrt5}\boldcdot\sqrt5$$

E:

$$\sqrt0\boldcdot\sqrt7$$

F:

$$\text-\sqrt5\boldcdot\sqrt5$$

G:

$$\sqrt5\boldcdot\sqrt7$$

### Problem 3

1. Where is the vertex of the graph that represents $$y=(x-3)^2 + 5$$?
2. Does the graph open up or down? Explain how you know.
(From Unit 6, Lesson 15.)

### Problem 4

Here are the solutions to some quadratic equations. Decide if the solutions are rational or irrational.

$$3 \pm \sqrt2$$

$$\sqrt9 \pm 1$$

$$\frac12 \pm \frac32$$

$$10 \pm 0.3$$

$$\frac{1 \pm \sqrt8}{2}$$

$$\text-7\pm\sqrt{\frac49}$$

### Problem 5

Find an example that shows that the statement is false.

1. An irrational number multiplied by an irrational number always makes an irrational product.
2. A rational number multiplied by an irrational number never gives a rational product.
3. Adding an irrational number to an irrational number always gives an irrational sum.

### Problem 6

Which equation is equivalent to $$x^2-\frac32x=\frac74$$ but has a perfect square on one side?

A:

$$x^2-\frac32x+3=\frac{19}{4}$$

B:

$$x^2-\frac32x+\frac34=\frac{10}{4}$$

C:

$$x^2-\frac32x+\frac94=\frac{16}{4}$$

D:

$$x^2-\frac32x+\frac94=\frac74$$

(From Unit 7, Lesson 13.)

### Problem 7

A student who used the quadratic formula to solve $$2x^2-8x=2$$ said that the solutions are $$x=2+\sqrt5$$ and $$x=2-\sqrt5$$

1. What equations can we graph to check those solutions? What features of the graph do we analyze?
2. How do we look for $$2+\sqrt5$$ and $$2-\sqrt5$$ on a graph?
(From Unit 7, Lesson 18.)

### Problem 8

Here are 4 graphs. Match each graph with a quadratic equation that it represents.

(From Unit 6, Lesson 15.)