Which expression is a factor of both [tex]x^2-9[/tex] and [tex]x^2+8 x+15[/tex]?
A. [tex](x+5)[/tex]
B. [tex](x+3)[/tex]
C. [tex](x-3)[/tex]
D. [tex](x-9)[/tex]
Answer (2)
We factorize both quadratic expressions and identify the common factor.
Factor x 2 − 9 as ( x − 3 ) ( x + 3 ) .
Factor x 2 + 8 x + 15 as ( x + 3 ) ( x + 5 ) .
Identify the common factor as ( x + 3 ) .
The common factor is ( x + 3 ) .
Explanation
Understanding the Problem We are given two quadratic expressions, x 2 − 9 and x 2 + 8 x + 15 , and we need to find a factor that is common to both. The possible factors are ( x + 5 ) , ( x + 3 ) , ( x − 3 ) , and ( x − 9 ) .
Factoring the First Expression First, let's factorize the expression x 2 − 9 . This is a difference of squares, so we can use the formula a 2 − b 2 = ( a − b ) ( a + b ) . In this case, a = x and b = 3 , so we have: x 2 − 9 = ( x − 3 ) ( x + 3 )
Factoring the Second Expression Next, let's factorize the expression x 2 + 8 x + 15 . We need to find two numbers that add up to 8 and multiply to 15. These numbers are 3 and 5. So, we can write: x 2 + 8 x + 15 = ( x + 3 ) ( x + 5 )
Identifying the Common Factor Now, let's compare the factors of both expressions:
x 2 − 9 has factors ( x − 3 ) and ( x + 3 ) .
x 2 + 8 x + 15 has factors ( x + 3 ) and ( x + 5 ) .
The common factor is ( x + 3 ) .
Final Answer Therefore, the expression that is a factor of both x 2 − 9 and x 2 + 8 x + 15 is ( x + 3 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model market behavior, and computer scientists use it to design efficient algorithms. Understanding how to factor quadratic expressions allows us to solve problems in a variety of fields.
The common factor of both expressions x 2 − 9 and x 2 + 8 x + 15 is ( x + 3 ) . Therefore, the correct answer is option B: ( x + 3 ) .
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