Expanding from Factored Form to Standard Form

9.10.2 Expanding from Factored Form to Standard Form

9.10.2 • Expanding from Factored Form to Standard Form

Warm Up

Recall learning how to identify and factor perfect square trinomials in Unit 6. Here are some perfect squares in factored and standard forms, and an expression showing how the two forms are related.

Study the first few examples in rows 1 and 2. Create a similar table to the one modeled below. Copy the table (or use a copy provided by your teacher) and then complete the missing numbers in rows 3 – 6 in the rest of the table.

Row	Factored Form		Standard Form
1	$(x + 4)^{2}$	$(1 x)^{2} + 2 (1 x) (4) + 4^{2}$	$x^{2} + 8 x + 16$
2	$(2 x + 5)^{2}$	$(2 x)^{2} + 2 (2 x) (5) + 5^{2}$	$4 x^{2} + 20 x + 25$
3	$(3 x - 4)^{2}$	$(3 x)^{2} + 2 (___x) (___) + (___)^{2}$	$9 x^{2} - 24 x + 16$
4	$(5 x +___)^{2}$	$(___x)^{2} + 2 (___x) (___) + (___)^{2}$	$25 x^{2} + 30 x +___$
5	$(k x + m)^{2}$	$(___x)^{2} + 2 (___x) (___) + (___)^{2}$	$(___)^{2} x^{2} +___x + (___)^{2}$

1. Enter the missing values from row 3.

Row	Factored Form		Standard Form
3	$(3 x - 4)^{2}$	$(3 x)^{2} + 2 (___x) (___) + (___)^{2}$ $(3 x)^{2} + 2 (\underset{―}{a .} x) (\underset{―}{b .}) + (\underset{―}{c .})^{2}$	$9 x^{2} - 24 x + 16$

Row

Factored Form

Standard Form

$(3 x - 4)^{2}$

$(3 x)^{2} + 2 (___x) (___) + (___)^{2}$

$(3 x)^{2} + 2 (\underset{―}{a .} x) (\underset{―}{b .}) + (\underset{―}{c .})^{2}$

$9 x^{2} - 24 x + 16$

a. What is the missing value of a?

b. What is the missing value of b?

-4.

c. What is the missing value of c?

-4.

2. Enter the missing values from row 4.

Row	Factored Form		Standard Form
4	$(5 x +___)^{2}$ $(5 x + \underset{―}{a .})^{2}$	$(___x)^{2} + 2 (___x) (___) + (___)^{2}$ $(\underset{―}{b .} x)^{2} + 2 (\underset{―}{c .} x) (\underset{―}{d .}) + (\underset{―}{e .})^{2}$	$25 x^{2} + 30 x +___$ $25 x^{2} + 30 x + \underset{―}{f .}$

Row

Factored Form

Standard Form

$(5 x +___)^{2}$

$(5 x + \underset{―}{a .})^{2}$

$(___x)^{2} + 2 (___x) (___) + (___)^{2}$

$(\underset{―}{b .} x)^{2} + 2 (\underset{―}{c .} x) (\underset{―}{d .}) + (\underset{―}{e .})^{2}$

$25 x^{2} + 30 x +___$

$25 x^{2} + 30 x + \underset{―}{f .}$

a. What is the missing value of a?

b. What is the missing value of b?

c. What is the missing value of c?

d. What is the missing value of d?

e. What is the missing value of e?

f. What is the missing value of f?

3. Enter the missing values from row 5.

Row	Factored Form		Standard Form
5	$(k x + m)^{2}$	$(___x)^{2} + 2 (___x) (___) + (___)^{2}$ $(\underset{―}{a .} x)^{2} + 2 (\underset{―}{b .} x) (\underset{―}{c .}) + (\underset{―}{d .})^{2}$	$(___)^{2} x^{2} +___x + (___)^{2}$ $(\underset{―}{e .})^{2} x^{2} + \underset{―}{f .} x + (\underset{―}{g .})^{2}$

Row

Factored Form

Standard Form

$(k x + m)^{2}$

$(___x)^{2} + 2 (___x) (___) + (___)^{2}$

$(\underset{―}{a .} x)^{2} + 2 (\underset{―}{b .} x) (\underset{―}{c .}) + (\underset{―}{d .})^{2}$

$(___)^{2} x^{2} +___x + (___)^{2}$

$(\underset{―}{e .})^{2} x^{2} + \underset{―}{f .} x + (\underset{―}{g .})^{2}$

a. What is the missing value of a?

$k$ .

b. What is the missing value of b?

$k$ .

c. What is the missing value of c?

$m$ .

d. What is the missing value of d?

$m$ .

e. What is the missing value of e?

$k$ .

f. What is the missing value of f?

$2 k m$ .

g. What is the missing value of g?

$m$ .

4. Look at the expression in the last row of the table. If $a x^{2} + b x + c$ is equivalent to $(k x + m)^{2}$ , how are $a$ , $b$ , and $c$ related to $k$ and $m$ ?

Compare your answer:

$a = k^{2}$ , $b = 2 k m$ , and $c = m^{2}$