Solved ?Factoring the Difference of Two Squares In Exercises ... - Free Printable

Problem: Factoring the Difference of Two Squares

The task is to factor each expression in Exercises 9–18 as the difference of two squares. The general formula for factoring the difference of two squares is:

\[
a^2 - b^2 = (a - b)(a + b)
\]

We will apply this formula to each given expression step by step.

---

Exercise 9: \( x^2 - 81 \)

1. Identify \( a \) and \( b \):
\[
x^2 - 81 = x^2 - 9^2
\]
Here, \( a = x \) and \( b = 9 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
x^2 - 81 = (x - 9)(x + 9)
\]

Answer:
\[
\boxed{(x - 9)(x + 9)}
\]

---

Exercise 10: \( x^2 - 64 \)

1. Identify \( a \) and \( b \):
\[
x^2 - 64 = x^2 - 8^2
\]
Here, \( a = x \) and \( b = 8 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
x^2 - 64 = (x - 8)(x + 8)
\]

Answer:
\[
\boxed{(x - 8)(x + 8)}
\]

---

Exercise 11: \( 25y^2 - 4 \)

1. Identify \( a \) and \( b \):
\[
25y^2 - 4 = (5y)^2 - 2^2
\]
Here, \( a = 5y \) and \( b = 2 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
25y^2 - 4 = (5y - 2)(5y + 2)
\]

Answer:
\[
\boxed{(5y - 2)(5y + 2)}
\]

---

Exercise 12: \( 4y^2 - 49 \)

1. Identify \( a \) and \( b \):
\[
4y^2 - 49 = (2y)^2 - 7^2
\]
Here, \( a = 2y \) and \( b = 7 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
4y^2 - 49 = (2y - 7)(2y + 7)
\]

Answer:
\[
\boxed{(2y - 7)(2y + 7)}
\]

---

Exercise 13: \( 64 - 9z^2 \)

1. Rewrite the expression to match the form \( a^2 - b^2 \):
\[
64 - 9z^2 = 8^2 - (3z)^2
\]
Here, \( a = 8 \) and \( b = 3z \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
64 - 9z^2 = (8 - 3z)(8 + 3z)
\]

Answer:
\[
\boxed{(8 - 3z)(8 + 3z)}
\]

---

Exercise 14: \( 81 - 36z^2 \)

1. Rewrite the expression to match the form \( a^2 - b^2 \):
\[
81 - 36z^2 = 9^2 - (6z)^2
\]
Here, \( a = 9 \) and \( b = 6z \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
81 - 36z^2 = (9 - 6z)(9 + 6z)
\]

Answer:
\[
\boxed{(9 - 6z)(9 + 6z)}
\]

---

Exercise 15: \( (x - 1)^2 - 4 \)

1. Identify \( a \) and \( b \):
\[
(x - 1)^2 - 4 = (x - 1)^2 - 2^2
\]
Here, \( a = x - 1 \) and \( b = 2 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
(x - 1)^2 - 4 = [(x - 1) - 2][(x - 1) + 2]
\]
Simplify each term:
\[
(x - 1) - 2 = x - 3 \quad \text{and} \quad (x - 1) + 2 = x + 1
\]
So,
\[
(x - 1)^2 - 4 = (x - 3)(x + 1)
\]

Answer:
\[
\boxed{(x - 3)(x + 1)}
\]

---

Exercise 16: \( 25 - (z + 5)^2 \)

1. Identify \( a \) and \( b \):
\[
25 - (z + 5)^2 = 5^2 - (z + 5)^2
\]
Here, \( a = 5 \) and \( b = z + 5 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
25 - (z + 5)^2 = [5 - (z + 5)][5 + (z + 5)]
\]
Simplify each term:
\[
5 - (z + 5) = 5 - z - 5 = -z \quad \text{and} \quad 5 + (z + 5) = 5 + z + 5 = z + 10
\]
So,
\[
25 - (z + 5)^2 = (-z)(z + 10)
\]
or equivalently,
\[
25 - (z + 5)^2 = -z(z + 10)
\]

Answer:
\[
\boxed{-z(z + 10)}
\]

---

Exercise 17: \( 81u^4 - 1 \)

1. Identify \( a \) and \( b \):
\[
81u^4 - 1 = (9u^2)^2 - 1^2
\]
Here, \( a = 9u^2 \) and \( b = 1 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
81u^4 - 1 = (9u^2 - 1)(9u^2 + 1)
\]

3. Notice that \( 9u^2 - 1 \) is itself a difference of squares:
\[
9u^2 - 1 = (3u)^2 - 1^2 = (3u - 1)(3u + 1)
\]
So,
\[
81u^4 - 1 = (3u - 1)(3u + 1)(9u^2 + 1)
\]

Answer:
\[
\boxed{(3u - 1)(3u + 1)(9u^2 + 1)}
\]

---

Exercise 18: \( x^4 - 16y^4 \)

1. Identify \( a \) and \( b \):
\[
x^4 - 16y^4 = (x^2)^2 - (4y^2)^2
\]
Here, \( a = x^2 \) and \( b = 4y^2 \).

2. Apply the formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
x^4 - 16y^4 = (x^2 - 4y^2)(x^2 + 4y^2)
\]

3. Notice that \( x^2 - 4y^2 \) is itself a difference of squares:
\[
x^2 - 4y^2 = (x)^2 - (2y)^2 = (x - 2y)(x + 2y)
\]
So,
\[
x^4 - 16y^4 = (x - 2y)(x + 2y)(x^2 + 4y^2)
\]

Answer:
\[
\boxed{(x - 2y)(x + 2y)(x^2 + 4y^2)}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
&9. \ (x - 9)(x + 9) \\
&10. \ (x - 8)(x + 8) \\
&11. \ (5y - 2)(5y + 2) \\
&12. \ (2y - 7)(2y + 7) \\
&13. \ (8 - 3z)(8 + 3z) \\
&14. \ (9 - 6z)(9 + 6z) \\
&15. \ (x - 3)(x + 1) \\
&16. \ -z(z + 10) \\
&17. \ (3u - 1)(3u + 1)(9u^2 + 1) \\
&18. \ (x - 2y)(x + 2y)(x^2 + 4y^2)
\end{aligned}
}
\]