Which is the correct simplified form of ( (x+y+z)^2-(x-y+z)^2 )?
Answer and explanation
Correct answer: (4y(x+z))
The expression contains two squares that differ only in the sign of the middle term. Group the terms that are unchanged by writing the expression as \\(A+y)^2-(A-y)^2\\), where \\(A=x+z\\). The identity \\( (A+B)^2-(A-B)^2=4AB\\) says that this difference is four times the product of the common part and the changing term. This avoids expanding every term separately.
Substituting \\(A=x+z\\) and \\(B=y\\) gives \\(4(x+z)y=4y(x+z)\\). Thus the correct simplified form is option A. Option B is exactly half the required value, option C keeps only an incomplete product, and option D does not represent the difference of these two squares. The grouping and identity also provide a quick check on the result.
Frequently asked questions
What is the correct answer to this question?
(4y(x+z))
Why is this the correct answer?
The expression contains two squares that differ only in the sign of the middle term. Group the terms that are unchanged by writing the expression as \\(A+y)^2-(A-y)^2\\), where \\(A=x+z\\). The identity \\( (A+B)^2-(A-B)^2=4AB\\) says that this difference is four times the product of the common part and the changing term. This avoids expanding every term separately.
Substituting \\(A=x+z\\) and \\(B=y\\) gives \\(4(x+z)y=4y(x+z)\\). Thus the correct simplified form is option A. Option B is exactly half the required value, option C keeps only an incomplete product, and option D does not represent the difference of these two squares. The grouping and identity also provide a quick check on the result.
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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