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If a=0.25, find the value of 100[1+2(2a)+4a2тАЛтИТ2a].
100
150
200
50
100
Observe that 1+4a+4a2 is the perfect square (1+2a)2. Thus, the expression simplifies to 100[(1+2a)тИТ2a], which is simply 100(1).
The variable a = 0.25. The expression given is 100[sqrt(1 + 2(2a) + 4a┬▓) - 2a].
(x+y)2=x2+2xy+y2
Observe that 1+4a+4a2 is the perfect square (1+2a)2. Thus, the expression simplifies to 100[(1+2a)тИТ2a], which is simply 100(1).
Students often attempt to substitute a=0.25 immediately into the radical expression, leading to complex decimal calculations and potential arithmetic errors.
Simplify the expression inside the square root
Recognize that the term 1+4a+4a2 can be rewritten as 12+2(1)(2a)+(2a)2.
1+4a+4a2=(1+2a)2
Substitute back into the main expression
Replace the square root part with its simplified form (1+2a) since the square root of (1+2a)2 is (1+2a).
100[(1+2a)2тАЛтИТ2a]=100[1+2aтИТ2a]
Calculate the final result
Cancel the 2a terms inside the bracket to get 1, then multiply by 100.
100[1]=100
A is correct because the algebraic expression simplifies to 100 regardless of the specific value of a, provided the terms under the root form a perfect square.
This type of simplification frequently appears in competitive exams to test your ability to recognize patterns (like a2+2ab+b2) before doing raw computation, which saves significant time.
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