M
Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 7

Factor Quadratics in Two Variables

Factor first- and second-degree polynomials in x and y completely.

Jump to section:

Key Ideas Easy Medium Hard Super Challenge
What to Remember

Treat y as part of the last term

x² + 5xy + 6y² is like x² + 5x + 6 with y attached to the second factor of each pair.

Multiply to the last, add to the middle

For x² + bxy + cy², find two numbers that multiply to c and add to b. Then write (x + py)(x + qy).

a > 1 still slides

Slide a onto the y² term, factor, divide, and keep y with the constants.

GCF first

2x² + 10xy + 12y² = 2(x² + 5xy + 6y²) = 2(x + 2y)(x + 3y).

Level 1

Easy

a = 1. Two-variable trinomials.

Example
Factor. x² + 5xy + 6y²

Two numbers that multiply to 6 and add to 5: 2 and 3.

(x + 2y)(x + 3y)

Check: x² + 3xy + 2xy + 6y² = x² + 5xy + 6y²

Answer: (x + 2y)(x + 3y)

1
Factor. x² + 7xy + 10y²
Show Answer

2 and 5

(x + 2y)(x + 5y)

Answer: (x + 2y)(x + 5y)

2
Factor. x² + 6xy + 8y²
Show Answer

2 and 4

(x + 2y)(x + 4y)

Answer: (x + 2y)(x + 4y)

3
Factor. x² − 9y²
Show Answer

(x + 3y)(x − 3y)

Answer: (x + 3y)(x − 3y)

4
Factor. x² + 8xy + 16y²
Show Answer

(x + 4y)²

Answer: (x + 4y)²

5
Factor. x² − xy − 6y²
Show Answer

−3 and 2

(x − 3y)(x + 2y)

Answer: (x − 3y)(x + 2y)

Level 2

Medium

GCF, or a > 1 with small numbers.

Example
Factor. 2x² + 10xy + 12y²

GCF = 2

2(x² + 5xy + 6y²)

2(x + 2y)(x + 3y)

Answer: 2(x + 2y)(x + 3y)

1
Factor. 3x² + 9xy + 6y²
Show Answer

3(x² + 3xy + 2y²)

3(x + y)(x + 2y)

Answer: 3(x + y)(x + 2y)

2
Factor. 2x² + 7xy + 3y²
Show Answer

Slide 2 onto 3y²: think x² + 7x + 6, then (x + 1)(x + 6).

Divide: (2x + 1)(x + 3y)

Check: 2x² + 6xy + xy + 3y² = 2x² + 7xy + 3y²

Answer: (2x + 1)(x + 3y)

3
Factor. x² − 4xy + 4y²
Show Answer

(x − 2y)²

Answer: (x − 2y)²

4
Factor. 4x² − 9y²
Show Answer

(2x + 3y)(2x − 3y)

Answer: (2x + 3y)(2x − 3y)

5
Factor. x² + xy − 12y²
Show Answer

4 and −3

(x + 4y)(x − 3y)

Answer: (x + 4y)(x − 3y)

Level 3

Hard

Slide with y, or two variables and a GCF.

Example
Factor. 3x² + 11xy + 6y²

Slide 3: x² + 11x + 18 → (x + 2)(x + 9).

Divide 2 and 9 by 3: (3x + 2y)(x + 3y)

Check: 3x² + 9xy + 2xy + 6y² = 3x² + 11xy + 6y²

Answer: (3x + 2y)(x + 3y)

1
Factor. 2x² − 5xy − 3y²
Show Answer

(2x + y)(x − 3y)

Answer: (2x + y)(x − 3y)

2
Factor completely. 4x² + 20xy + 24y²
Show Answer

4(x² + 5xy + 6y²)

4(x + 2y)(x + 3y)

Answer: 4(x + 2y)(x + 3y)

3
Factor. 6x² + 7xy − 3y²
Show Answer

(3x − y)(2x + 3y)

Answer: (3x − y)(2x + 3y)

4
Factor. 9x² − 12xy + 4y²
Show Answer

(3x − 2y)²

Answer: (3x − 2y)²

5
Factor. x² − 16y²
Show Answer

(x + 4y)(x − 4y)

Answer: (x + 4y)(x − 4y)

Super Challenge

Super Challenge

1
Show that (x + 2y)(x + 3y) = x² + 5xy + 6y² two ways: distribute, and an area rectangle with sides x + 2y and x + 3y.
Show Answer

x² + 3xy + 2xy + 6y²

The four regions are x², 3xy, 2xy, and 6y².

Answer: x² + 5xy + 6y²

2
Factor x² + (a + b)xy + aby².
Show Answer

(x + ay)(x + by)

Answer: (x + ay)(x + by)

3
A student factors 2x² + 5xy + 3y² as (2x + 3y)(x + y). Check, then correct if needed.
Show Answer

(2x + 3y)(x + y) = 2x² + 2xy + 3xy + 3y² = 2x² + 5xy + 3y². Correct.

Answer: (2x + 3y)(x + y)

Need help factoring two-variable polynomials?

Book a Session with Amy