Mathematics is often described as the study of patterns, so it is not surprising
that area models have many patterns. You saw one important pattern in
Lesson 4.1.1 (Casey’s pattern from problem 4-4). Today you will continue to
use patterns while you develop a method to factor trinomial expressions.

4-12. Examine the area model shown at
right.
a.
Review what you learned in Lesson 4.1.1 by
writing the area of the rectangle at right as a
sum and as a product.
(5x – 2)(2x – 7) = 10x2 − 39x + 14
b.
Does this area model fit Casey’s pattern for
diagonals? State the pattern and justify your
answer.
The products of the diagonals are equal;
−35x · −4x = 10x2 · 14 = 140x2
4.1.2 Factoring with Area Models
November 16, 2015
Objectives
 CO: SWBAT
develop an algorithm to
factor quadratic expressions without
using algebra tiles.
 LO: SWBAT
explain to a partner
how to factor quadratic expressions.
4-13. FACTORING
QUADRATIC
EXPRESSIONS
a.

A polynomial in the form ax2 + bx + c, with a ≠ 0, is called a quadratic
expression in standard form. To develop a method for factoring
quadratic expressions without using algebra tiles, you will first model
how to factor with algebra tiles, and then look for connections within
an area model.
Using algebra tiles, factor 2x2 + 5x + 3; that is,
use the tiles to build a rectangle, and then
write its area as a product.
(2x + 3)(x + 1)
b.
When you factor using tiles, you need to
determine how to arrange the tiles to form a
rectangle. Using an area model to factor
requires a slightly different process.
Miguel wants to use an area model to factor
3x2 + 10x + 8. He knows that 3x2 and 8 go
into the rectangle along one of the diagonals,
as shown at right. Finish the rectangle by
deciding how to split and place the remaining
term. Then write the area as a product.
(3x + 4)(x + 2)
Algebra tiles
4-13. FACTORING QUADRATIC
EXPRESSIONS
c.
Kelly wants to determine a method for factoring
2x2 + 7x + 6. She remembers Casey’s pattern for
diagonals. She places 2x2 and 6 into the rectangle
in the locations shown at right. Without actually
factoring yet, what do you know about the missing
two parts of the area model?
Their sum is 7x, and their product is 12x2
d.
To complete Kelly’s area model, create and solve a
Diamond Problem using the information from part
(c).
The product 12x2 should be placed at the top of the
diamond problem, 7x at the bottom, and the terms 3x and
4x should be in the middle.
e.
Use your results from the Diamond Problem to
complete the area model for 2x2 + 7x + 6, and then
write the area as a product of factors.
(2x + 3)(x + 2)
So, then…?
1.
2.
3.
4.
5.
6.
7.
Place the x2-term and the constant term of the quadratic
expression in opposite corners of the area model.
Determine the product of the x2-term and the constant
term. Place this product in the top of a Diamond Problem.
Place the x-term from the expression in the bottom (sum) of
a Diamond Problem.
Solve the Diamond Problem by identifying two terms whose
product and sum match those in the diamond.
Place the solutions from the Diamond Problem into the two
remaining corners of the area model.
Determine the dimensions of the area model.
Write the area as the product of the dimensions.
4-14. Factoring with an area model is especially convenient when algebra tiles are not
available or when the number of necessary tiles becomes too large to manage. Using a
Diamond Problem helps avoid guessing and checking, which can at times be
challenging. Use the process from problem 4-13 to factor 6x2 + 17x + 12.

1.
2.
3.
4.
5.
(2x + 3)(3x + 4)
Place x2 & # in box
Set up diamond
Solve diamond
Find dimensions
Write product
4-16. Use the process you developed in problems 4-13 and 4-14 to
factor the following quadratic expressions, if possible. If a quadratic
cannot be factored, justify your conclusion.
a.
b.
c.
d.
x2 + 9x + 18
4x2 + 17x – 15
4x2 – 8x + 3
3x2 + 5x – 3
a.
b.
c.
d.
(x + 3)(x + 6)
(4x − 3)(x + 5)
(2x − 3)(2x − 1)
not factorable