Factor completely: 36×2 – 49
The Correct Answer and Explanation is :
To factor the expression ( 36x^2 – 49 ), we recognize that it is a difference of squares. A difference of squares is any expression in the form ( a^2 – b^2 ), which factors as ( (a – b)(a + b) ).
Step-by-Step Solution:
- Recognize the structure of the expression:
The given expression ( 36x^2 – 49 ) is a difference of squares because both terms are perfect squares:
[
36x^2 = (6x)^2 \quad \text{and} \quad 49 = 7^2
] - Apply the difference of squares formula:
The formula is:
[
a^2 – b^2 = (a – b)(a + b)
]
Here, ( a = 6x ) and ( b = 7 ). Using the formula, we can factor the expression:
[
36x^2 – 49 = (6x – 7)(6x + 7)
]
Thus, the factored form of ( 36x^2 – 49 ) is:
[
(6x – 7)(6x + 7)
]
Explanation (300 words):
In algebra, factoring expressions is an essential technique used to simplify polynomials and solve equations. One important pattern to recognize is the difference of squares, which occurs when a binomial is the result of subtracting one square number from another. The general form of a difference of squares is ( a^2 – b^2 ), and it factors neatly into the product of two binomials: ( (a – b)(a + b) ).
In this problem, ( 36x^2 – 49 ), both terms are perfect squares. The term ( 36x^2 ) is the square of ( 6x ), and ( 49 ) is the square of ( 7 ). Once we recognize this, we can apply the difference of squares formula.
Factoring using this pattern simplifies the expression into two linear binomials: ( (6x – 7)(6x + 7) ). This technique is especially useful because it turns a quadratic expression into a product of simpler terms. Factoring helps in solving equations, simplifying expressions, and understanding relationships between variables.
In summary, the factorization process involved recognizing that the expression fits the difference of squares form and applying the appropriate factoring formula. This is a powerful algebraic tool that helps break down more complex polynomials into simpler parts.