Simplify: ${(3 a - 2)}^{2} - {(2 a - 3)}^{2}$

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Solution

Using the identity: $(x - y)^{2} = x^{2} + y^{2} - 2 x y$ .

Given: ${(3 a - 2)}^{2} - {(2 a - 3)}^{2}$ , we apply the above identity as follows:

${(3 a - 2)}^{2} - {(2 a - 3)}^{2}$

$= [{(3 a)}^{2} + {(2)}^{2} - (2 \times 3 a \times 2)] - [{(2 a)}^{2} + {(3)}^{2} - (2 \times 2 a \times 3)]$

$= (9 a^{2} + 4 - 12 a) - (4 a^{2} + 9 - 12 a)$

$= 9 a^{2} + 4 - 12 a - 4 a^{2} - 9 + 12 a$

$= 5 a^{2} - 5$

$= 5 (a^{2} - 1)$

Hence, ${(3 a - 2)}^{2} - {(2 a - 3)}^{2} = 5 (a^{2} - 1)$ .

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