Match the following identities:

a^{2} - 9

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a^{2} + 4 a + 4

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a^{2} - 2 a + 1

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2 (a^{2} + 1)

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Solution

We know that $(x + y)^{2} = x^{2} + y^{2} + 2 x y$
So, $(a + 2)^{2} = a^{2} + 2^{2} + 2 \times a \times 2$
$= a^{2} + 4 + 4 a$

We know that $(x - y)^{2} = x^{2} + y^{2} - 2 x y$
So, $(a - 1)^{2} = a^{2} + 1^{2} - 2 \times a \times 1$
$= a^{2} + 1 - 2 a$

We know that $(x + y) (x - y) = x^{2} - y^{2}$
So, $(a + 3) (a - 3) = a^{2} - 3^{2} = a^{2} - 9$

Similarly, $(a + 1)^{2} + (a - 1)^{2}$
$= a^{2} + 1 + 2 a + a^{2} + 1 - 2 a$
$= 2 a^{2} + 2 = 2 (a^{2} + 1)$

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