Factor $16 x^2+40 x+25$
A. $(4 x+5)(-4 x-5)$
B. $(4 x-5)^2$
C. $(4 x+5)^2$
D. $(-4 x+5)^2$
Answer (2)
Recognize the quadratic expression as a potential perfect square trinomial.
Identify p and q such that p 2 = 16 and q 2 = 25 , so p = 4 and q = 5 .
Verify that the middle term matches 2 pq x , which is 2 ( 4 ) ( 5 ) x = 40 x .
Conclude that the factored form is ( 4 x + 5 ) 2 , so the answer is ( 4 x + 5 ) 2 .
Explanation
Problem Analysis We are asked to factor the quadratic expression 16 x 2 + 40 x + 25 . We need to identify which of the given options is the correct factorization.
Recognizing the Quadratic Form The given expression is a quadratic of the form a x 2 + b x + c , where a = 16 , b = 40 , and c = 25 . We can check if this is a perfect square trinomial.
Checking for Perfect Square 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 . In our case, we have p 2 = 16 and q 2 = 25 , which gives us p = 4 and q = 5 .
Verifying the Middle Term Now we check the middle term. We have 2 pq = 2 ( 4 ) ( 5 ) = 40 , which matches the middle term 40 x in the given expression. Therefore, the expression is of the form ( 4 x + 5 ) 2 .
Expanding to Verify To verify, we expand ( 4 x + 5 ) 2 : ( 4 x + 5 ) 2 = ( 4 x ) 2 + 2 ( 4 x ) ( 5 ) + 5 2 = 16 x 2 + 40 x + 25 This matches the original expression.
Conclusion Therefore, the factored form of 16 x 2 + 40 x + 25 is ( 4 x + 5 ) 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 analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Factoring helps simplify complex equations, making them easier to solve and understand. Imagine you're designing a rectangular garden with an area represented by the expression 16 x 2 + 40 x + 25 . By factoring this expression to ( 4 x + 5 ) 2 , you realize the garden is a square with side length 4 x + 5 . This allows you to easily determine the amount of fencing needed.
The factored form of the expression 16 x 2 + 40 x + 25 is ( 4 x + 5 ) 2 . Therefore, the correct answer is option C. This indicates that the expression is a perfect square trinomial.
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