Difference of Squares

Show your work for each problem.

1.
Which expression is a difference of squares?
1. a) $x^2 - 25$
2. b) $x^2 - 5$
3. c) $x - 25$
4. d) $x^2 + 25$
Answer: $x^2 - 25$
$x^2 - 25$ is a difference of squares because $x^2$ and $25 = 5^2$ are both perfect squares with a minus sign between them. The sum $x^2 + 25$ cannot be factored, $5$ is not a perfect square, and $x - 25$ has no squares.
2.
Fill in the boxes to factor $x^2 - 9$ : $x^2 - 9 = (x + \square)(x - \square)$ . What number goes in both boxes?
Answer: 3
$x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)$ using the formula $a^2 - b^2 = (a+b)(a-b)$ where $a = x$ and $b = 3$ .
3.
Factor $x^2 - 49$ step by step.
Answer: (x+7)(x-7)
1. Is $x^2$ a perfect square? yes
2. What is the square root of 49? 7
3. Which is the factored form of $x^2 - 49$ ? $(x + 7)(x - 7)$
4.
What is the factored form of $y^2 - 100$ ?
1. a) $(y - 10)(y - 10)$
2. b) $(y + 100)(y - 100)$
3. c) $(y + 10)(y - 10)$
4. d) $(y + 50)(y - 50)$
Answer: $(y + 10)(y - 10)$
$y^2 - 100 = y^2 - 10^2 = (y + 10)(y - 10)$ . Since $\sqrt{100} = 10$ , we use $b = 10$ in the formula.
5.
Factor $a^2 - 36$ step by step.
Answer: (a+6)(a-6)
1. What is the square root of $a^2$ ? a
2. What is the square root of 36? 6
3. Which is the factored form of $a^2 - 36$ ? $(a + 6)(a - 6)$
6.
Fill in the boxes to factor $4x^2 - 25$ : $4x^2 - 25 = (\square + 5)(\square - 5)$ . What goes in both boxes?
Answer: 2x
$4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5)$ . Here $a = 2x$ and $b = 5$ .
7.
Which of the following can NOT be factored as a difference of squares?
1. a) $9y^2 - 1$
2. b) $x^2 - 16$
3. c) $x^2 + 16$
4. d) $25 - z^2$
Answer: $x^2 + 16$
$x^2 + 16$ is a SUM of squares, not a difference. The sum of squares cannot be factored using real numbers. The others are all differences: $x^2 - 16 = (x+4)(x-4)$ , $9y^2 - 1 = (3y+1)(3y-1)$ , $25 - z^2 = (5+z)(5-z)$ .
8.
Factor $16m^2 - 81$ step by step.
Answer: (4m+9)(4m-9)
1. Write $16m^2$ as a perfect square: $(?)^2$ 4m
2. Write $81$ as a perfect square: $(?)^2$ 9
3. Apply the formula $(a+b)(a-b)$ : $(4m + 9)(4m - 9)$
9.
Fill in the boxes to factor $49 - y^2$ : $49 - y^2 = (\square + y)(\square - y)$ . What number goes in both boxes?
Answer: 7
$49 - y^2 = 7^2 - y^2 = (7 + y)(7 - y)$ . The order is reversed because the constant (49) comes first.
10.
Factor $25a^2 - 4b^2$ step by step.
Answer: (5a+2b)(5a-2b)
1. Write $25a^2$ as $(?)^2$ 5a
2. Write $4b^2$ as $(?)^2$ 2b
3. Apply the formula: $(5a + 2b)(5a - 2b)$
11.
Use mental math: What is $47 \times 53$ ?
1. a)2500
2. b)2401
3. c)2509
4. d)2491
Answer: 2491
$47 \times 53 = (50 - 3)(50 + 3) = 50^2 - 3^2 = 2500 - 9 = 2491$
12.
Factor $x^4 - 16$ completely. The result has three factors: $(x^2 + 4)(x + \square)(x - \square)$ . What number goes in both boxes?
Answer: 2
$x^4 - 16 = (x^2)^2 - 4^2 = (x^2 + 4)(x^2 - 4)$ . Now $x^2 - 4$ is another difference of squares: $x^2 - 4 = (x+2)(x-2)$ . The sum $x^2 + 4$ cannot be factored. Final answer: $(x^2 + 4)(x + 2)(x - 2)$
13.
Factor completely: $81 - x^4$
Answer: (9+x^2)(3+x)(3-x)
1. $81 = 9^2$ . Write $x^4$ as a square: $x^4 = (?)^2$ x^2
2. Apply the formula with $a = 9$ and $b = x^2$ . Which two factors do you get? $(9 + x^2)(9 - x^2)$
3. Can $(9 - x^2)$ be factored further? yes
4. $9 - x^2 = (\square + x)(\square - x)$ . What number goes in both boxes? 3
5. Which is the complete factorization of $81 - x^4$ ? $(9 + x^2)(3 + x)(3 - x)$
14.
Factor $16x^4 - 1$ completely. How many factors does the result have?
1. a)Cannot be factored
2. b)3 factors
3. c)4 factors
4. d)2 factors
Answer: 3 factors
$16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 + 1)(4x^2 - 1)$ . The factor $4x^2 - 1 = (2x)^2 - 1^2 = (2x + 1)(2x - 1)$ is again a difference of squares, while the sum $4x^2 + 1$ cannot be factored over the real numbers. Complete factorization: $(4x^2 + 1)(2x + 1)(2x - 1)$ , so 3 factors.
15.
Use mental math: Calculate $98 \times 102$
Answer: 9996
$98 \times 102 = (100 - 2)(100 + 2) = 100^2 - 2^2 = 10000 - 4 = 9996$
16.
Factor: $36x^4 - 49y^2$
Answer: (6x^2+7y)(6x^2-7y)
1. Write $36x^4$ as a perfect square: $(?)^2$ 6x^2
2. Write $49y^2$ as a perfect square: $(?)^2$ 7y
3. Apply the difference of squares formula: $(6x^2 + 7y)(6x^2 - 7y)$
17.
Fill in the boxes to factor $x^2 - 1$ : $x^2 - 1 = (x + \square)(x - \square)$ . What number goes in both boxes?
Answer: 1
$x^2 - 1 = x^2 - 1^2 = (x + 1)(x - 1)$
18.
A square garden has side length $(x + 5)$ meters. In the middle is a square pond with side length $(x - 5)$ meters. Find the area of the garden around the pond.
Answer: 20x
1. The area is $a^2 - b^2$ . What is $a$ , the side of the garden? x+5
2. What is $b$ , the side of the pond? x-5
3. Calculate $a + b$ : 2x
4. Calculate $a - b$ : 10
5. Multiply to get the final answer: 20x