What is the factored form of \( x^2 - 9 \)?

What is the factored form of \( x^2 - 9 \)?

• A. \( (x - 9)(x + 1) \)
• B. \( (x - 3)(x + 3) \)
• C. \( (x - 6)(x + 6) \)
• D. \( (x - 4)(x + 4) \)

Answer: (B)

The expression \( x^2 - 9 \) is recognized as a difference of squares, which is a special factoring technique that applies when you have a quadratic expression in the form \( a^2 - b^2 \). In this case, \( x^2 \) is \( a^2 \) and \( 9 \) is \( b^2 \) since \( 9 \) can be expressed as \( 3^2 \). The difference of squares can be factored using the formula: \[ a^2 - b^2 = (a - b)(a + b) \] Applying this to our expression, we identify \( a = x \) and \( b = 3 \). Therefore, we have: \[ x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3) \] This gives us the factored form of \( x^2 - 9 \) as \( (x - 3)(x + 3) \). This reasoning leads us to conclude why the second option is correct. The other options do not represent the correct factors for \( x^2 - 9 \).

The expression ( x^2 - 9 ) is recognized as a difference of squares, which is a special factoring technique that applies when you have a quadratic expression in the form ( a^2 - b^2 ). In this case, ( x^2 ) is ( a^2 ) and ( 9 ) is ( b^2 ) since ( 9 ) can be expressed as ( 3^2 ).

The difference of squares can be factored using the formula:

[

a^2 - b^2 = (a - b)(a + b)

]

Applying this to our expression, we identify ( a = x ) and ( b = 3 ). Therefore, we have:

[

x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)

]

This gives us the factored form of ( x^2 - 9 ) as ( (x - 3)(x + 3) ).

This reasoning leads us to conclude why the second option is correct. The other options do not represent the correct factors for ( x^2 - 9 ).