Factor completely x2 − 36.
The Correct Answer and Explanation is:
To factor the expression ( x^2 – 36 ), we can recognize that this is a difference of squares. A difference of squares follows the pattern:
[
a^2 – b^2 = (a – b)(a + b)
]
Step-by-Step Process:
- Identify the terms:
- The expression is ( x^2 – 36 ), which can be written as:
[
x^2 – 6^2
] - Here, ( a^2 = x^2 ) and ( b^2 = 6^2 ).
- Apply the difference of squares formula:
- Using the formula ( a^2 – b^2 = (a – b)(a + b) ), we can factor the expression as:
[
(x – 6)(x + 6)
]
This is the factored form of ( x^2 – 36 ).
Why This Works:
The reason this works is due to the difference of squares identity, which is based on the algebraic property that for any two terms ( a ) and ( b ):
[
a^2 – b^2 = (a – b)(a + b)
]
This identity states that the difference between two perfect squares can be factored into the product of two binomials, one representing the difference and the other the sum of the terms.
Verifying the Factorization:
To confirm that the factorization is correct, we can expand ( (x – 6)(x + 6) ) using the distributive property (also known as the FOIL method for binomials):
[
(x – 6)(x + 6) = x(x + 6) – 6(x + 6)
]
[
= x^2 + 6x – 6x – 36
]
[
= x^2 – 36
]
This matches the original expression, confirming that the factorization is correct.
Thus, the complete factorization of ( x^2 – 36 ) is:
[
(x – 6)(x + 6)
]
Conclusion:
By recognizing the difference of squares pattern and applying the formula ( a^2 – b^2 = (a – b)(a + b) ), we can easily factor the expression ( x^2 – 36 ) into ( (x – 6)(x + 6) ). This is a common algebraic technique used to simplify expressions and solve equations.