Factor $9 x^2+12 x+4$
A. $(-3 x+2)^2$
B. $(3 x-2)^2$
C. $(3 x+2)(-3 x-2)$
D. $(3 x+2)^2$
Answer (2)
Recognize the quadratic expression as a potential perfect square trinomial.
Verify that the first and last terms are perfect squares: 9 x 2 = ( 3 x ) 2 and 4 = 2 2 .
Check if the middle term fits the perfect square pattern: 12 x = 2 ( 3 x ) ( 2 ) .
Factor the expression as ( 3 x + 2 ) 2 , which matches one of the given options. The final answer is ( 3 x + 2 ) 2 .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 9 x 2 + 12 x + 4 . We need to identify which of the given options is the correct factorization.
Recognizing the Form We can recognize that the given quadratic expression is in the form of a x 2 + b x + c , where a = 9 , b = 12 , and c = 4 . We can check if this quadratic expression is a perfect square trinomial. A perfect square trinomial has the form ( p x + q ) 2 = p 2 x 2 + 2 pq x + q 2 or ( p x − q ) 2 = p 2 x 2 − 2 pq x + q 2 .
Checking for Perfect Square In our expression, we have 9 x 2 = ( 3 x ) 2 and 4 = 2 2 . So, we can rewrite the expression as ( 3 x ) 2 + 12 x + 2 2 . Now, we need to check if the middle term 12 x is equal to 2 ( 3 x ) ( 2 ) .
Verifying the Middle Term Calculating 2 ( 3 x ) ( 2 ) , we get 12 x . Since the middle term of the quadratic expression is indeed 12 x , we can conclude that the given expression is a perfect square trinomial.
Factoring the Expression Therefore, we can factor the expression as 9 x 2 + 12 x + 4 = ( 3 x + 2 ) 2 .
Identifying the Correct Option Comparing the factored expression ( 3 x + 2 ) 2 with the given options, we find that it matches the option ( 3 x + 2 ) 2 . Therefore, the correct factorization is ( 3 x + 2 ) 2 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and solve problems related to stress and strain. Imagine you are designing a rectangular garden with an area represented by the expression 9 x 2 + 12 x + 4 . By factoring this expression to ( 3 x + 2 ) 2 , you determine that the garden is a square with side length 3 x + 2 . This allows you to plan the layout and fencing efficiently.
The quadratic expression 9 x 2 + 12 x + 4 factors to ( 3 x + 2 ) 2 , which matches option D. This confirms that it's a perfect square trinomial. The breakdown involves verifying each term as a perfect square and checking the relation of the middle term.
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