Which visual model can immediately connect (x^2+2xy+y^2-z^2) with ((x+y-z)(x+y+z))?
Answer and explanation
Correct answer: Removing a square of side (z) from a square of side (x+y)
The expression \(x^2+2xy+y^2\) is the expansion of \((x+y)^2\). Therefore the full expression becomes \((x+y)^2-z^2\), which is a difference of two squares. The corresponding visual model starts with a large square of side \(x+y\), whose area is \((x+y)^2\), and removes a smaller square of side z, whose area is \(z^2\). Applying the difference-of-squares identity gives \((x+y-z)(x+y+z)\).
Thus option A is correct. A square of side \(x-y\) would produce a different perfect square, and two isolated rectangles cannot represent the complete expression. A full square of side \(x+y+z\) would include extra terms rather than subtracting \(z^2\). The useful visual sequence is therefore: form the \(x+y\) square, remove the z square, and factor the remaining difference.
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What is the correct answer to this question?
Removing a square of side (z) from a square of side (x+y)
Why is this the correct answer?
The expression \(x^2+2xy+y^2\) is the expansion of \((x+y)^2\). Therefore the full expression becomes \((x+y)^2-z^2\), which is a difference of two squares. The corresponding visual model starts with a large square of side \(x+y\), whose area is \((x+y)^2\), and removes a smaller square of side z, whose area is \(z^2\). Applying the difference-of-squares identity gives \((x+y-z)(x+y+z)\).
Thus option A is correct. A square of side \(x-y\) would produce a different perfect square, and two isolated rectangles cannot represent the complete expression. A full square of side \(x+y+z\) would include extra terms rather than subtracting \(z^2\). The useful visual sequence is therefore: form the \(x+y\) square, remove the z square, and factor the remaining difference.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Visual models of identities.
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