What is the completely factored form of $x^2-16xy+64y^2$?
A. $xy(x-16+64y)$
B. $xy(x+16+64y)$
C. $(x-8y)(x-8y)$
D. $(x+8y)(x+8y)$
Answer (2)
Recognize the quadratic expression as a perfect square trinomial.
Identify the terms a = x and b = 8 y in the expression.
Apply the perfect square trinomial formula: ( a − b ) 2 = ( x − 8 y ) 2 .
The completely factored form is ( x − 8 y ) ( x − 8 y ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 16 x y + 64 y 2 and asked to factor it completely.
Recognizing the Pattern We can recognize this expression as a perfect square trinomial. A perfect square trinomial has the form a 2 − 2 ab + b 2 , which factors to ( a − b ) 2 .
Identifying a and b In our expression, we can identify a as x and b as 8 y . Let's check if this fits the pattern:
a 2 = x 2 b 2 = ( 8 y ) 2 = 64 y 2 2 ab = 2 ( x ) ( 8 y ) = 16 x y
Since our expression is x 2 − 16 x y + 64 y 2 , it matches the form a 2 − 2 ab + b 2 exactly.
Factoring the Expression Now we can factor the expression using the formula ( a − b ) 2 . Substituting a = x and b = 8 y , we get:
( x − 8 y ) 2 = ( x − 8 y ) ( x − 8 y )
Final Answer Therefore, the completely factored form of x 2 − 16 x y + 64 y 2 is ( x − 8 y ) ( x − 8 y ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra. It's used in many real-world applications, such as optimizing areas and volumes, modeling projectile motion, and solving engineering problems. For example, if you're designing a rectangular garden with an area represented by x 2 − 16 x y + 64 y 2 , factoring it into ( x − 8 y ) ( x − 8 y ) helps you determine the dimensions of the garden in terms of x and y. This allows you to plan the layout efficiently and make the most of your space.
The total charge delivered by the device is 450 coulombs over 30 seconds. To find the number of electrons, we divide this charge by the charge of a single electron, yielding approximately 2.81 x 10^21 electrons. Thus, about 2.81 trillion electrons flow through the device during the specified time period.
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