Answer :
To factor a difference of squares, you can use the formula:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Let's apply this formula to each of the given expressions:
a. [tex]\( x^2 - 16 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 16 \)[/tex] is a square of 4 ([tex]\( 4^2 \)[/tex]).
2. Apply the difference of squares formula:
[tex]\[ x^2 - 16 = (x + 4)(x - 4) \][/tex]
b. [tex]\( x^2 - 81 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 81 \)[/tex] is a square of 9 ([tex]\( 9^2 \)[/tex]).
2. Apply the formula:
[tex]\[ x^2 - 81 = (x + 9)(x - 9) \][/tex]
c. [tex]\( x^2 - 64 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 64 \)[/tex] is a square of 8 ([tex]\( 8^2 \)[/tex]).
2. Apply the formula:
[tex]\[ x^2 - 64 = (x + 8)(x - 8) \][/tex]
d. [tex]\( 4x^2 - 49 \)[/tex]
1. Identify the squares:
- [tex]\( 4x^2 \)[/tex] is a square of [tex]\( 2x \)[/tex] ([tex]\( (2x)^2 \)[/tex]).
- [tex]\( 49 \)[/tex] is a square of 7 ([tex]\( 7^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 4x^2 - 49 = (2x + 7)(2x - 7) \][/tex]
e. [tex]\( 25x^2 - 4 \)[/tex]
1. Identify the squares:
- [tex]\( 25x^2 \)[/tex] is a square of [tex]\( 5x \)[/tex] ([tex]\( (5x)^2 \)[/tex]).
- [tex]\( 4 \)[/tex] is a square of 2 ([tex]\( 2^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 25x^2 - 4 = (5x + 2)(5x - 2) \][/tex]
f. [tex]\( 9x^2 - 25 \)[/tex]
1. Identify the squares:
- [tex]\( 9x^2 \)[/tex] is a square of [tex]\( 3x \)[/tex] ([tex]\( (3x)^2 \)[/tex]).
- [tex]\( 25 \)[/tex] is a square of 5 ([tex]\( 5^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 9x^2 - 25 = (3x + 5)(3x - 5) \][/tex]
h. [tex]\( 4x^2 - 49y^2 \)[/tex]
1. Identify the squares:
- [tex]\( 4x^2 \)[/tex] is a square of [tex]\( 2x \)[/tex] ([tex]\( (2x)^2 \)[/tex]).
- [tex]\( 49y^2 \)[/tex] is a square of [tex]\( 7y \)[/tex] ([tex]\( (7y)^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 4x^2 - 49y^2 = (2x + 7y)(2x - 7y) \][/tex]
Each expression is factored correctly using the difference of squares method.
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Let's apply this formula to each of the given expressions:
a. [tex]\( x^2 - 16 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 16 \)[/tex] is a square of 4 ([tex]\( 4^2 \)[/tex]).
2. Apply the difference of squares formula:
[tex]\[ x^2 - 16 = (x + 4)(x - 4) \][/tex]
b. [tex]\( x^2 - 81 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 81 \)[/tex] is a square of 9 ([tex]\( 9^2 \)[/tex]).
2. Apply the formula:
[tex]\[ x^2 - 81 = (x + 9)(x - 9) \][/tex]
c. [tex]\( x^2 - 64 \)[/tex]
1. Identify the squares:
- [tex]\( x^2 \)[/tex] is already a square ([tex]\( x^2 \)[/tex]).
- [tex]\( 64 \)[/tex] is a square of 8 ([tex]\( 8^2 \)[/tex]).
2. Apply the formula:
[tex]\[ x^2 - 64 = (x + 8)(x - 8) \][/tex]
d. [tex]\( 4x^2 - 49 \)[/tex]
1. Identify the squares:
- [tex]\( 4x^2 \)[/tex] is a square of [tex]\( 2x \)[/tex] ([tex]\( (2x)^2 \)[/tex]).
- [tex]\( 49 \)[/tex] is a square of 7 ([tex]\( 7^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 4x^2 - 49 = (2x + 7)(2x - 7) \][/tex]
e. [tex]\( 25x^2 - 4 \)[/tex]
1. Identify the squares:
- [tex]\( 25x^2 \)[/tex] is a square of [tex]\( 5x \)[/tex] ([tex]\( (5x)^2 \)[/tex]).
- [tex]\( 4 \)[/tex] is a square of 2 ([tex]\( 2^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 25x^2 - 4 = (5x + 2)(5x - 2) \][/tex]
f. [tex]\( 9x^2 - 25 \)[/tex]
1. Identify the squares:
- [tex]\( 9x^2 \)[/tex] is a square of [tex]\( 3x \)[/tex] ([tex]\( (3x)^2 \)[/tex]).
- [tex]\( 25 \)[/tex] is a square of 5 ([tex]\( 5^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 9x^2 - 25 = (3x + 5)(3x - 5) \][/tex]
h. [tex]\( 4x^2 - 49y^2 \)[/tex]
1. Identify the squares:
- [tex]\( 4x^2 \)[/tex] is a square of [tex]\( 2x \)[/tex] ([tex]\( (2x)^2 \)[/tex]).
- [tex]\( 49y^2 \)[/tex] is a square of [tex]\( 7y \)[/tex] ([tex]\( (7y)^2 \)[/tex]).
2. Apply the formula:
[tex]\[ 4x^2 - 49y^2 = (2x + 7y)(2x - 7y) \][/tex]
Each expression is factored correctly using the difference of squares method.
