Multiply using the Product of Conjugates Pattern.
$(2 x+5 y)(2 x-5 y) =$
Answer (1)
Identify a and b : a = 2 x and b = 5 y .
Apply the product of conjugates pattern: ( a + b ) ( a − b ) = a 2 − b 2 .
Substitute a and b into the formula: ( 2 x ) 2 − ( 5 y ) 2 .
Simplify the expression to get the final answer: 4 x 2 − 25 y 2 .
Explanation
Understanding the problem We are asked to multiply two binomials using the product of conjugates pattern. The two binomials are ( 2 x + 5 y ) and ( 2 x − 5 y ) . The product of conjugates pattern is ( a + b ) ( a − b ) = a 2 − b 2 .
Identifying a and b In the given expression ( 2 x + 5 y ) ( 2 x − 5 y ) , we can identify a = 2 x and b = 5 y .
Applying the pattern Applying the product of conjugates pattern, we have ( a + b ) ( a − b ) = a 2 − b 2 . Substituting a = 2 x and b = 5 y into the formula, we get ( 2 x ) 2 − ( 5 y ) 2 .
Simplifying the expression Simplifying the expression, we have ( 2 x ) 2 = 4 x 2 and ( 5 y ) 2 = 25 y 2 . Therefore, the final result is 4 x 2 − 25 y 2 .
Final result The product of ( 2 x + 5 y ) ( 2 x − 5 y ) using the product of conjugates pattern is 4 x 2 − 25 y 2 .
Examples
The product of conjugates pattern is useful in many areas of mathematics and physics. For example, it can be used to simplify expressions involving square roots or to solve equations. Imagine you are designing a rectangular garden where the length is ( 2 x + 5 y ) meters and the width is ( 2 x − 5 y ) meters. Using the product of conjugates, you can easily find the area of the garden, which is ( 2 x + 5 y ) ( 2 x − 5 y ) = 4 x 2 − 25 y 2 square meters. This pattern allows for quick calculation of areas or other similar quantities without needing to perform the full multiplication.
