# 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 are always certain kinds of numbers.

### Problem 1

Match each expression to an equivalent expression.

### 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.

\(\sqrt4\boldcdot\sqrt5\)

\(\sqrt4\boldcdot\sqrt4\)

\(\sqrt7\boldcdot\sqrt7\)

\(\frac{1}{\sqrt5}\boldcdot\sqrt5\)

\(\sqrt0\boldcdot\sqrt7\)

\(\text-\sqrt5\boldcdot\sqrt5\)

\(\sqrt5\boldcdot\sqrt7\)

### Problem 3

- Where is the vertex of the graph that represents \(y=(x-3)^2 + 5\)?
- Does the graph open up or down? Explain how you know.

### 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.

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

### Problem 6

Which equation is equivalent to \(x^2-3x=\frac74\) but has a perfect square on one side?

\(x^2-3x+3=\frac{19}{4}\)

\(x^2-3x+\frac34=\frac{10}{4}\)

\(x^2-3x+\frac94=\frac{16}{4}\)

\(x^2-3x+\frac94=\frac74\)

### Problem 7

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

- What equations can we graph to check those solutions? What features of the graph do we analyze?
- How do we look for \(2+\sqrt5\) and \(2-\sqrt5\) on a graph?

### Problem 8

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