# Lesson 6

Squares and Square Roots

• Let’s compare equations with squares and square roots.

### 6.1: Math Talk: Four Squares

Find the solutions of each equation mentally.

$$x^2 = 4$$

$$x^2 = 2$$

$$x^2 = 0$$

$$x^2 = \text{-}1$$

### 6.2: Finding Square Roots

Clare was adding $$\sqrt{4}$$ and $$\sqrt{9}$$, and at first she wrote $$\sqrt{4} + \sqrt{9} = 2+3$$. But then she remembered that 2 and -2 both square to make 4, and that 3 and -3 both square to make 9. She wrote down all the possible combinations:

2 + 3 = 5
2 + (-3) = -1
(-2) + 3 = 1
(-2) + (-3) = -5

Then she wondered, “Which of these are the same as $$\sqrt{4} + \sqrt{9}$$? All of them? Or only some? Or just one?”

1. How many solutions are there to each equation?
1. $$x^3=8$$
2. $$y^3=\text -1$$
3. $$z^4 = 16$$
4. $$w^4 = \text -81$$
2. Write a rule to determine how many solutions there are to the equation $$x^n=m$$ where $$n$$ and $$m$$ are non-zero integers.

### 6.3: One Solution or Two?

1. The graph of $$b=\sqrt{a}$$ is shown.

1. Complete the table with the exact values and label the corresponding points on the graph with the exact values.
 $$a$$ $$\sqrt{a}$$ 1 4 9 12 16 20
2. Label the point on the graph that shows the solution to $$\sqrt{a} = 4$$.
3. Label the point on the graph that shows the solution to $$\sqrt{a} = 5$$.
4. Label the point on the graph that shows the solution to $$\sqrt{a} = \sqrt{5}$$.
2. The graph of $$t = s^2$$ is shown.

1. Label the point(s) on the graph that show(s) the solution(s) to $$s^2 = 25$$.
2. Label the point(s) on the graph that show(s) the solution(s) to $$\sqrt{t} = 5$$.
3. Label the point(s) on the graph that show(s) the solution(s) to $$s^2 = 5$$.

### Summary

The symbol $$\sqrt{11}$$ represents the positive square root of 11. If we want to represent the negative square root, we write $$\text{-} \sqrt{11}$$.

The equation $$x^2 = 11$$ has two solutions, because $$\sqrt{11}^2=11$$, and also$$\left(\text{-}\sqrt{11}\right)^2=11$$.

The equation $$\sqrt{x} = 11$$ only has one solution, namely 121.

The equation $$\sqrt{x} = \sqrt{11}$$ only has one solution, namely 11.

The equation $$\sqrt{x} = \text{-}11$$ doesn’t have any solutions, because the left side is positive and the right side is negative, which is impossible, because a positive number cannot equal a negative number.