What is the simplified form of ( (6x+1)^2-(6x-1)^2 )?
Answer and explanation
Correct answer: (24x)
The expression is a difference of two squares. The identity \\(a^2-b^2=4ab\\) is not the general identity; rather, when the squares are \\((a+b)^2-(a-b)^2\\), their difference is \\(4ab\\). Here, take \\(a=6x\\) and \\(b=1\\). Therefore, \\((6x+1)^2-(6x-1)^2=4(6x)(1)=24x\\). Thus, option B is correct.
Expanding also confirms the result. We have \\((6x+1)^2=36x^2+12x+1\\) and \\((6x-1)^2=36x^2-12x+1\\). Subtracting the second expression cancels both \\(36x^2\\) and the constant terms, while the linear terms give \\(12x-(-12x)=24x\\). Option A, \\(12x\\), misses half of the difference. Options C and D incorrectly retain square or constant terms that cancel. The complete term \\(6x\\), not merely 6, must be used for a.
Frequently asked questions
What is the correct answer to this question?
(24x)
Why is this the correct answer?
The expression is a difference of two squares. The identity \\(a^2-b^2=4ab\\) is not the general identity; rather, when the squares are \\((a+b)^2-(a-b)^2\\), their difference is \\(4ab\\). Here, take \\(a=6x\\) and \\(b=1\\). Therefore, \\((6x+1)^2-(6x-1)^2=4(6x)(1)=24x\\). Thus, option B is correct.
Expanding also confirms the result. We have \\((6x+1)^2=36x^2+12x+1\\) and \\((6x-1)^2=36x^2-12x+1\\). Subtracting the second expression cancels both \\(36x^2\\) and the constant terms, while the linear terms give \\(12x-(-12x)=24x\\). Option A, \\(12x\\), misses half of the difference. Options C and D incorrectly retain square or constant terms that cancel. The complete term \\(6x\\), not merely 6, must be used for a.
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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