What is the simplified form of ( (8x+1)^2-(8x-1)^2 )?
Answer and explanation
Correct answer: (32x)
This expression has the form \\((a+b)^2-(a-b)^2\\), whose simplification is \\(4ab\\). Set \\(a=8x\\) and \\(b=1\\). Then \\(4ab=4(8x)(1)=32x\\), so option B is correct. The important point is that a represents the complete term \\(8x\\), not just the numerical coefficient 8. This is why the answer is linear in x rather than a constant or a squared expression.
Expanding gives the same confirmation: \\((8x+1)^2=64x^2+16x+1\\), while \\((8x-1)^2=64x^2-16x+1\\). On subtraction, the two \\(64x^2\\) terms cancel, as do the constant 1 terms. The remaining part is \\(16x-(-16x)=32x\\). Option A is only half of the correct difference; options C and D retain terms that should cancel. Hence the second choice follows exactly from the identity.
Frequently asked questions
What is the correct answer to this question?
(32x)
Why is this the correct answer?
This expression has the form \\((a+b)^2-(a-b)^2\\), whose simplification is \\(4ab\\). Set \\(a=8x\\) and \\(b=1\\). Then \\(4ab=4(8x)(1)=32x\\), so option B is correct. The important point is that a represents the complete term \\(8x\\), not just the numerical coefficient 8. This is why the answer is linear in x rather than a constant or a squared expression.
Expanding gives the same confirmation: \\((8x+1)^2=64x^2+16x+1\\), while \\((8x-1)^2=64x^2-16x+1\\). On subtraction, the two \\(64x^2\\) terms cancel, as do the constant 1 terms. The remaining part is \\(16x-(-16x)=32x\\). Option A is only half of the correct difference; options C and D retain terms that should cancel. Hence the second choice follows exactly from the identity.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Algebraic identities.
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