Difference of Two Squares Worksheet | Cazoom Maths Worksheets - Free Printable

Problem: Factorize the given expressions using the difference of two squares method.

The difference of two squares formula is:
\[
a^2 - b^2 = (a - b)(a + b)
\]

We will solve each part step by step.

---

Section A: Factorize using the difference of two squares

#### 1. \( x^2 - 25 \)
- Recognize that \( 25 = 5^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5)
\]
- Answer: \((x - 5)(x + 5)\)

#### 2. \( x^2 - 49 \)
- Recognize that \( 49 = 7^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 49 = x^2 - 7^2 = (x - 7)(x + 7)
\]
- Answer: \((x - 7)(x + 7)\)

#### 3. \( x^2 - 100 \)
- Recognize that \( 100 = 10^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 100 = x^2 - 10^2 = (x - 10)(x + 10)
\]
- Answer: \((x - 10)(x + 10)\)

#### 4. \( x^2 - 225 \)
- Recognize that \( 225 = 15^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 225 = x^2 - 15^2 = (x - 15)(x + 15)
\]
- Answer: \((x - 15)(x + 15)\)

#### 5. \( 2x^2 - 32 \)
- First, factor out the common factor \( 2 \):
\[
2x^2 - 32 = 2(x^2 - 16)
\]
- Recognize that \( 16 = 4^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4)
\]
- Substitute back:
\[
2x^2 - 32 = 2(x - 4)(x + 4)
\]
- Answer: \(2(x - 4)(x + 4)\)

#### 6. \( 5x^2 - 45 \)
- First, factor out the common factor \( 5 \):
\[
5x^2 - 45 = 5(x^2 - 9)
\]
- Recognize that \( 9 = 3^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)
\]
- Substitute back:
\[
5x^2 - 45 = 5(x - 3)(x + 3)
\]
- Answer: \(5(x - 3)(x + 3)\)

#### 7. \( 4x^2 - 144 \)
- First, recognize that \( 4x^2 = (2x)^2 \) and \( 144 = 12^2 \).
- Apply the difference of two squares formula:
\[
4x^2 - 144 = (2x)^2 - 12^2 = (2x - 12)(2x + 12)
\]
- Simplify if possible (though not necessary here):
\[
(2x - 12)(2x + 12)
\]
- Answer: \((2x - 12)(2x + 12)\)

#### 8. \( 7x^2 - 567 \)
- First, factor out the common factor \( 7 \):
\[
7x^2 - 567 = 7(x^2 - 81)
\]
- Recognize that \( 81 = 9^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 81 = x^2 - 9^2 = (x - 9)(x + 9)
\]
- Substitute back:
\[
7x^2 - 567 = 7(x - 9)(x + 9)
\]
- Answer: \(7(x - 9)(x + 9)\)

---

Section B: Factorize using the difference of two squares

#### 1. \( 4a^2 - 9 \)
- Recognize that \( 4a^2 = (2a)^2 \) and \( 9 = 3^2 \).
- Apply the difference of two squares formula:
\[
4a^2 - 9 = (2a)^2 - 3^2 = (2a - 3)(2a + 3)
\]
- Answer: \((2a - 3)(2a + 3)\)

#### 2. \( 36s^2 - 121 \)
- Recognize that \( 36s^2 = (6s)^2 \) and \( 121 = 11^2 \).
- Apply the difference of two squares formula:
\[
36s^2 - 121 = (6s)^2 - 11^2 = (6s - 11)(6s + 11)
\]
- Answer: \((6s - 11)(6s + 11)\)

#### 3. \( 64 - p^2 \)
- Recognize that \( 64 = 8^2 \).
- Apply the difference of two squares formula:
\[
64 - p^2 = 8^2 - p^2 = (8 - p)(8 + p)
\]
- Answer: \((8 - p)(8 + p)\)

#### 4. \( 25 - 16c^2 \)
- Recognize that \( 25 = 5^2 \) and \( 16c^2 = (4c)^2 \).
- Apply the difference of two squares formula:
\[
25 - 16c^2 = 5^2 - (4c)^2 = (5 - 4c)(5 + 4c)
\]
- Answer: \((5 - 4c)(5 + 4c)\)

#### 5. \( x^2 - 9y^2 \)
- Recognize that \( 9y^2 = (3y)^2 \).
- Apply the difference of two squares formula:
\[
x^2 - 9y^2 = x^2 - (3y)^2 = (x - 3y)(x + 3y)
\]
- Answer: \((x - 3y)(x + 3y)\)

#### 6. \( 63 - 7q^2 \)
- First, factor out the common factor \( 7 \):
\[
63 - 7q^2 = 7(9 - q^2)
\]
- Recognize that \( 9 = 3^2 \).
- Apply the difference of two squares formula:
\[
9 - q^2 = 3^2 - q^2 = (3 - q)(3 + q)
\]
- Substitute back:
\[
63 - 7q^2 = 7(3 - q)(3 + q)
\]
- Answer: \(7(3 - q)(3 + q)\)

#### 7. \( 196 - 4v^2 \)
- Recognize that \( 196 = 14^2 \) and \( 4v^2 = (2v)^2 \).
- Apply the difference of two squares formula:
\[
196 - 4v^2 = 14^2 - (2v)^2 = (14 - 2v)(14 + 2v)
\]
- Simplify if possible (though not necessary here):
\[
(14 - 2v)(14 + 2v)
\]
- Answer: \((14 - 2v)(14 + 2v)\)

#### 8. \( 4x^2 - 121y^2 \)
- Recognize that \( 4x^2 = (2x)^2 \) and \( 121y^2 = (11y)^2 \).
- Apply the difference of two squares formula:
\[
4x^2 - 121y^2 = (2x)^2 - (11y)^2 = (2x - 11y)(2x + 11y)
\]
- Answer: \((2x - 11y)(2x + 11y)\)

---

Extension: Factorize \( 2a^3b - 8ab^3 \)

#### Step 1: Factor out the greatest common factor (GCF)
- The GCF of \( 2a^3b \) and \( 8ab^3 \) is \( 2ab \).
- Factor out \( 2ab \):
\[
2a^3b - 8ab^3 = 2ab(a^2 - 4b^2)
\]

#### Step 2: Factorize the remaining expression \( a^2 - 4b^2 \)
- Recognize that \( 4b^2 = (2b)^2 \).
- Apply the difference of two squares formula:
\[
a^2 - 4b^2 = a^2 - (2b)^2 = (a - 2b)(a + 2b)
\]

#### Step 3: Substitute back
- Combine the results:
\[
2a^3b - 8ab^3 = 2ab(a - 2b)(a + 2b)
\]

- Answer: \(2ab(a - 2b)(a + 2b)\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{Section A:} \\
&1. (x - 5)(x + 5) \\
&2. (x - 7)(x + 7) \\
&3. (x - 10)(x + 10) \\
&4. (x - 15)(x + 15) \\
&5. 2(x - 4)(x + 4) \\
&6. 5(x - 3)(x + 3) \\
&7. (2x - 12)(2x + 12) \\
&8. 7(x - 9)(x + 9) \\
\\
&\text{Section B:} \\
&1. (2a - 3)(2a + 3) \\
&2. (6s - 11)(6s + 11) \\
&3. (8 - p)(8 + p) \\
&4. (5 - 4c)(5 + 4c) \\
&5. (x - 3y)(x + 3y) \\
&6. 7(3 - q)(3 + q) \\
&7. (14 - 2v)(14 + 2v) \\
&8. (2x - 11y)(2x + 11y) \\
\\
&\text{Extension:} \\
&2ab(a - 2b)(a + 2b)
\end{aligned}
}
\]