Factor $9 x^2-30 x+25$.
$(3 x+5)^2$
$(3 x+5)(-3 x-5)$
$(3 x-5)^2$
$(-3 x-5)^2$
Answer (2)
Recognize the quadratic expression as a potential perfect square trinomial.
Express the first and last terms as squares: 9 x 2 = ( 3 x ) 2 and 25 = 5 2 .
Verify that the middle term fits the perfect square trinomial pattern: − 30 x = − 2 ( 3 x ) ( 5 ) .
Factor the quadratic expression as ( 3 x − 5 ) 2 . The final answer is ( 3 x − 5 ) 2 .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 9 x 2 − 30 x + 25 . 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 of the form a x 2 + b x + c , where a = 9 , b = − 30 , and c = 25 . We can check if this quadratic expression is a perfect square trinomial. A perfect square trinomial is of 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 .
Expressing as a Square In our case, we have 9 x 2 − 30 x + 25 . We can write 9 x 2 = ( 3 x ) 2 and 25 = 5 2 . So, we can try to express the quadratic as ( 3 x − 5 ) 2 .
Expanding the Square Expanding ( 3 x − 5 ) 2 , we get ( 3 x − 5 ) ( 3 x − 5 ) = ( 3 x ) 2 − 2 ( 3 x ) ( 5 ) + 5 2 = 9 x 2 − 30 x + 25 .
Factored Form Since ( 3 x − 5 ) 2 = 9 x 2 − 30 x + 25 , the factored form of the quadratic expression is ( 3 x − 5 ) 2 .
Alternative Forms Alternatively, we can also write ( − 3 x + 5 ) 2 = ( − ( 3 x − 5 ) ) 2 = ( 3 x − 5 ) 2 . Also, ( − 3 x − 5 ) 2 = ( 3 x + 5 ) 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 calculate stress and strain. Imagine you're designing a rectangular garden with an area represented by the quadratic expression 9 x 2 − 30 x + 25 . By factoring this expression to ( 3 x − 5 ) 2 , you determine that the garden is a square with side length ( 3 x − 5 ) . This allows you to plan the layout and fencing efficiently.
The quadratic expression 9 x 2 − 30 x + 25 can be factored as ( 3 x − 5 ) 2 because it is a perfect square trinomial. This matches the structure of a 2 − 2 ab + b 2 where a = 3 x and b = 5 . Therefore, the correct option is ( 3 x − 5 ) 2 .
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