Which factor do $9 x^2-12 x+4$ and $9 x^2-4$ have in common?
A. $3 x+4$
B. $3 x+2$
C. $9 x^2$
D. $3 x-2$
Answer (1)
Factorize the first expression: 9 x 2 − 12 x + 4 = ( 3 x − 2 ) 2 .
Factorize the second expression: 9 x 2 − 4 = ( 3 x − 2 ) ( 3 x + 2 ) .
Identify the common factor: ( 3 x − 2 ) .
The common factor is 3 x − 2 .
Explanation
Problem Analysis We are given two quadratic expressions, 9 x 2 − 12 x + 4 and 9 x 2 − 4 , and we need to find their common factor from the given options.
Factorizing the first expression First, let's factorize the expression 9 x 2 − 12 x + 4 . This is a quadratic expression of the form a x 2 + b x + c . We can rewrite it as ( 3 x ) 2 − 2 ( 3 x ) ( 2 ) + ( 2 ) 2 . This is a perfect square trinomial, which can be factored as ( 3 x − 2 ) 2 or ( 3 x − 2 ) ( 3 x − 2 ) .
Factorizing the second expression Now, let's factorize the second expression 9 x 2 − 4 . This is a difference of squares, which can be written as ( 3 x ) 2 − ( 2 ) 2 . Using the difference of squares formula, a 2 − b 2 = ( a − b ) ( a + b ) , we can factorize it as ( 3 x − 2 ) ( 3 x + 2 ) .
Identifying the common factor Comparing the factors of both expressions, we have:
9 x 2 − 12 x + 4 = ( 3 x − 2 ) ( 3 x − 2 ) 9 x 2 − 4 = ( 3 x − 2 ) ( 3 x + 2 )
The common factor is ( 3 x − 2 ) .
Final Answer From the given options, the common factor is 3 x − 2 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to design structures and analyze stress distribution. Architects use factoring to create symmetrical and aesthetically pleasing designs. Financial analysts use factoring to model and predict market trends. By understanding factoring, students can develop critical thinking and problem-solving skills that are applicable in many fields.
