# Lesson 20

Rational and Irrational Solutions

### Problem 1

Decide whether each number is rational or irrational.

• 10
• $$\frac45$$
• $$\sqrt4$$
• $$\sqrt{10}$$
• -3
• $$\sqrt{\frac{25}{4}}$$
• $$\sqrt{0.6}$$

### Solution

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### Problem 2

Here are the solutions to some quadratic equations. Select all solutions that are rational.

A:

$$5 \pm 2$$

B:

$$\sqrt4 \pm 1$$

C:

$$\frac12 \pm 3$$

D:

$$10 \pm \sqrt3$$

E:

$$\pm \sqrt{25}$$

F:

$$1 \pm \sqrt2$$

### Solution

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### Problem 3

Solve each equation. Then, determine if the solutions are rational or irrational.

1. $$(x+1)^2 = 4$$
2. $$(x-5)^2 = 36$$
3. $$(x+3)^2 = 11$$
4. $$(x-4)^2 = 6$$

### Solution

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### Problem 4

Here is a graph of the equation $$y=81(x-3)^2-4$$.

1. Based on the graph, what are the solutions to the equation $$81(x-3)^2=4$$?

2. Can you tell whether they are rational or irrational? Explain how you know.
3. Solve the equation using a different method and say whether the solutions are rational or irrational. Explain or show your reasoning.

### Solution

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### Problem 5

Match each equation to an equivalent equation with a perfect square on one side.

### Solution

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(From Unit 7, Lesson 13.)

### Problem 6

To derive the quadratic formula, we can multiply $$ax^2+bx+c=0$$ by an expression so that the coefficient of $$x^2$$ is a perfect square and the coefficient of $$x$$ is an even number.

1. Which expression, $$a$$, $$2a$$, or $$4a$$, would you multiply $$ax^2+bx+c=0$$ by to get started deriving the quadratic formula?
2. What does the equation $$ax^2+bx+c=0$$ look like when you multiply both sides by your answer?

### Solution

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(From Unit 7, Lesson 19.)

### Problem 7

Here is a graph that represents $$y=x^2$$.

On the same coordinate plane, sketch and label the graph that represents each equation:

1. $$y=\text-x^2-4$$
2. $$y=2x^2+4$$

### Solution

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(From Unit 6, Lesson 12.)

### Problem 8

Which quadratic expression is in vertex form?

A:

$$x^2-6x+8$$

B:

$$(x-6)^2+3$$

C:

$$(x-3)(x-6)$$

D:

$$(8-x)x$$

### Solution

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(From Unit 6, Lesson 15.)

### Problem 9

Function $$f$$ is defined by the expression $$\frac{5}{x-2}$$.

1. Evaluate $$f(12)$$.
2. Explain why $$f(2)$$ is undefined.
3. Give a possible domain for $$f$$.

### Solution

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(From Unit 4, Lesson 10.)