How would the expression $\left(x^2+4\right)\left(y^2+4\right)$ be rewritten using Two Squares?
A. $(2 x y+4)^2+(2 x+2 y)^2$
B. $(x y-2)^2-(2 x+2 y)^2$
C. $(x y-4)^2+(2 x+2 y)^2$
D. $(x y-16)^2-(4 x+4 y)^2
Answer (2)
The expression ( x 2 + 4 ) ( y 2 + 4 ) can be rewritten using the Two Squares identity as ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 . The correct option is C. This follows from applying the identity with appropriate substitutions.
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Apply the Two Squares identity: ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 2 .
Substitute a = x , b = 2 , c = y , and d = 2 into the identity.
Simplify the expression to get ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 .
The rewritten expression is ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 .
Explanation
Understanding the Problem We are given the expression ( x 2 + 4 ) ( y 2 + 4 ) and asked to rewrite it using the Two Squares identity. The Two Squares identity states that ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 2 = ( a c + b d ) 2 + ( a d − b c ) 2 . We want to rewrite ( x 2 + 4 ) ( y 2 + 4 ) in the form ( A ) 2 + ( B ) 2 and determine which of the given options is correct.
Applying the Two Squares Identity We can rewrite the expression as ( x 2 + 2 2 ) ( y 2 + 2 2 ) . Now, we apply the Two Squares identity with a = x , b = 2 , c = y , and d = 2 . Using the identity ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 2 , we have ( x 2 + 2 2 ) ( y 2 + 2 2 ) = ( x y − 2 ⋅ 2 ) 2 + ( x ⋅ 2 + 2 ⋅ y ) 2 = ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 .
Identifying the Correct Option Comparing the result ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 with the given options, we see that option C matches our result.
Final Answer Therefore, the expression ( x 2 + 4 ) ( y 2 + 4 ) can be rewritten as ( x y − 4 ) 2 + ( 2 x + 2 y ) 2 . The correct option is C.
Examples
The Two Squares identity is useful in various areas of mathematics, including number theory and algebra. For example, it can be used to factorize certain types of expressions or to simplify calculations involving complex numbers. In physics, this identity can appear when dealing with energy conservation in systems with quadratic terms. Imagine you're designing a suspension bridge, and you need to calculate the tension and compression forces. The Two Squares identity can help simplify these calculations, ensuring the bridge is stable and safe.
