# Lesson 11

What are Perfect Squares?

### Problem 1

Select all the expressions that are perfect squares.

A:

$$(x+5)(x+5)$$

B:

$$(\text- 9 + c)(c-9)$$

C:

$$(y-10)(10-y)$$

D:

$$(a+3)(3+a)$$

E:

$$(2x-1)(2x+1)$$

F:

$$(4-3x)(3-4x)$$

G:

$$(a+b)(b+a)$$

### Problem 2

Each diagram represents the square of an expression or a perfect square.

$$(n+7)^2$$

$$n$$       $$7$$
$$n$$    $$n^2$$ $$7n$$
$$7$$   $$7n$$ $$7^2$$

$$(5-m)^2$$

$$5$$ $$\text-m$$
$$5$$ $$5^2$$ $$5(\text-m)$$
$$\text-m$$ $$5(\text-m)$$ $$(\text-m)^2$$

$$(h+\frac13)^2$$

$$h$$       $$\frac13$$
$$h$$
$$\frac13$$
1. Complete the cells in the last table.
2. How are the contents of the three diagrams alike? This diagram represents $$(\text{term_1}+\text{term_2})^2$$. Describe your observations about cells 1, 2, 3, and 4.
term_1 term_2
term_1 cell 1 cell 2
term_2 cell 3 cell 4

3. Rewrite the perfect-square expressions $$(n+7)^2$$, $$(5-m)^2$$, and $$(h+\frac13)^2$$ in standard form: $$ax^2+bx+c$$.
4. How are the $$ax^2$$, $$bx$$, and $$c$$ of a perfect square in standard form related to the two terms in $$(\text{term_1}+\text{term_2})^2$$?

### Problem 3

Solve each equation.

1. $$(x - 1)^2 = 4$$
2. $$(x + 5)^2 = 81$$
3. $$(x - 2)^2 = 0$$
4. $$(x + 11)^2 = 121$$
5. $$(x - 7)^2 = \frac{64}{49}$$

### Problem 4

Explain or show why the product of a sum and a difference, such as $$(2x+1)(2x-1)$$, has no linear term when written in standard form.

### Solution

(From Unit 7, Lesson 8.)

### Problem 5

To solve the equation $$(x+3)^2=4$$, Han first expanded the squared expression. Here is his incomplete work:

\begin{align}(x+3)^2&=4\\ (x+3)(x+3)&=4\\ x^2+3x+3x+9&=4\\ x^2+6x+9&=4 \end{align}

1. Complete Han’s work and solve the equation.
2. Jada saw the equation $$(x+3)^2=4$$ and thought, “There are two numbers, 2 and -2, that equal 4 when squared. This means $$x+3$$ is either 2 or it is -2. I can find the values of $$x$$ from there.”

Use Jada’s reasoning to solve the equation.

3. Can Jada use her reasoning to solve $$(x+3)(x-3)=5$$? Explain your reasoning.