Multiply using the Binomial Squares Pattern.
[tex]$(2 a-1)^2=2 a^2$[/tex]
Answer (2)
Recall the binomial squares pattern: ( x − y ) 2 = x 2 − 2 x y + y 2 .
Apply the pattern to ( 2 a − 1 ) 2 : ( 2 a − 1 ) 2 = ( 2 a ) 2 − 2 ( 2 a ) ( 1 ) + ( 1 ) 2 .
Simplify the expression: ( 2 a ) 2 − 2 ( 2 a ) ( 1 ) + ( 1 ) 2 = 4 a 2 − 4 a + 1 .
The correct expansion is 4 a 2 − 4 a + 1 .
Explanation
Understanding the Problem We are asked to multiply ( 2 a − 1 ) 2 using the binomial squares pattern. The given equation ( 2 a − 1 ) 2 = 2 a 2 is incorrect, and we need to find the correct expansion.
Recalling the Binomial Squares Pattern The binomial squares pattern states that for any x and y , ( x − y ) 2 = x 2 − 2 x y + y 2 . We will use this pattern to expand ( 2 a − 1 ) 2 .
Applying the Pattern Let x = 2 a and y = 1 . Applying the binomial squares pattern, we have: ( 2 a − 1 ) 2 = ( 2 a ) 2 − 2 ( 2 a ) ( 1 ) + ( 1 ) 2
Simplifying the Expression Now, we simplify the expression: ( 2 a ) 2 − 2 ( 2 a ) ( 1 ) + ( 1 ) 2 = 4 a 2 − 4 a + 1
Identifying the Error Therefore, the correct expansion of ( 2 a − 1 ) 2 is 4 a 2 − 4 a + 1 . The given equation ( 2 a − 1 ) 2 = 2 a 2 is incorrect.
Final Answer The correct expansion of ( 2 a − 1 ) 2 using the binomial squares pattern is 4 a 2 − 4 a + 1 .
Examples
The binomial squares pattern is useful in various fields, such as physics and engineering, where squared expressions frequently appear. For example, when calculating the kinetic energy of an object, which is given by the formula K E = 2 1 m v 2 , if the velocity v is expressed as a binomial, say v = ( a − b ) , then v 2 = ( a − b ) 2 can be expanded using the binomial squares pattern to simplify the energy calculation. This pattern also finds applications in financial mathematics, such as calculating compound interest or analyzing investment growth, where squared terms often arise in formulas.
Using the Binomial Squares Pattern, the expression ( 2 a − 1 ) 2 expands to 4 a 2 − 4 a + 1 . This is derived by applying the formula ( x − y ) 2 = x 2 − 2 x y + y 2 . Thus, the correct expansion is confirmed as 4 a 2 − 4 a + 1 .
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