Difference of Squares

Show your work for each problem.

1.
Which expression is a difference of squares?
1. a)
2. b)
3. c)
4. d)
Answer:
is a difference of squares because and are both perfect squares with a minus sign between them. The sum cannot be factored, is not a perfect square, and has no squares.
2.
Factor: . Write your answer in the form where is a number.
Answer: (x+3)(x-3)
using the formula where and .
3.
Factor step by step.
Answer: (x+7)(x-7)
1. Is a perfect square? yes
2. What is the square root of 49? 7
3. What is ? (x+7)(x-7)
4.
What is the factored form of ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Since , we use in the formula.
5.
Factor step by step.
Answer: (a+6)(a-6)
1. What is the square root of ? a
2. What is the square root of 36? 6
3. Write the factored form using : (a+6)(a-6)
6.
Factor:
Answer: (2x+5)(2x-5)
. Here and .
7.
Which of the following can NOT be factored as a difference of squares?
1. a)
2. b)
3. c)
4. d)
Answer:
is a SUM of squares, not a difference. The sum of squares cannot be factored using real numbers. The others are all differences: , , .
8.
Factor step by step.
Answer: (4m+9)(4m-9)
1. Write as a perfect square: 4m
2. Write as a perfect square: 9
3. Apply the formula : (4m+9)(4m-9)
9.
Factor:
Answer: (7+y)(7-y)
. The order is reversed because the constant (49) comes first.
10.
Factor step by step.
Answer: (5a+2b)(5a-2b)
1. Write as 5a
2. Write as 2b
3. Apply the formula: (5a+2b)(5a-2b)
11.
Use mental math: What is ?
1. a)2500
2. b)2401
3. c)2509
4. d)2491
Answer: 2491
12.
Factor completely:
Answer: (x^2+4)(x+2)(x-2)
. Now is another difference of squares: . The sum cannot be factored. Final answer:
13.
Factor completely:
Answer: (9+x^2)(3+x)(3-x)
1. Write as a difference of squares: 9^2 - (x^2)^2
2. Apply the formula to get two factors: (9+x^2)(9-x^2)
3. Can be factored further? yes
4. Factor : (3+x)(3-x)
5. Write the complete factorization: (9+x^2)(3+x)(3-x)
14.
Factor completely. How many factors does it have?
1. a)Cannot be factored
2. b)3 factors
3. c)4 factors
4. d)2 factors
Answer: 4 factors
(difference of squares). Now factor each cubic using sum/difference of cubes: and . The complete factorization is , giving 4 irreducible polynomial factors.
15.
Use mental math: Calculate
Answer: 9996
16.
Factor:
Answer: (6x^2+7y)(6x^2-7y)
1. Write as a perfect square: 6x^2
2. Write as a perfect square: 7y
3. Apply the difference of squares formula: (6x^2+7y)(6x^2-7y)
17.
Factor:
Answer: (x+1)(x-1)
18.
A square garden has side length meters. A smaller square path inside has side length meters. Find the area of the garden excluding the path.
Answer: 20x
1. Write the expression for the area difference: (x+5)^2 - (x-5)^2
2. Identify and for the difference of squares: a=x+5, b=x-5
3. Calculate : 2x
4. Calculate : 10
5. Multiply to get the final answer: 20x