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Question
Simplify : (2.5p − 1.5q)2 − (1.5p − 2.5q)2
Solution
To simplify, we will proceed as follows:
\[\left( 2 . 5p - 1 . 5q \right)^2 - \left( 1 . 5p - 2 . 5q \right)^2 \]
\[ = \left( 2 . 5p \right)^2 + \left( 1 . 5q \right)^2 - 2\left( 2 . 5p \right)\left( 1 . 5q \right) - \left[ \left( 1 . 5p \right)^2 + \left( 2 . 5q \right)^2 - 2\left( 1 . 5p \right)\left( 2 . 5q \right) \right] \]
\[ = \left( 2 . 5p \right)^2 + \left( 1 . 5q \right)^2 - 2\left( 2 . 5p \right)\left( 1 . 5q \right) - \left( 1 . 5p \right)^2 - \left( 2 . 5q \right)^2 + 2\left( 1 . 5p \right)\left( 2 . 5q \right)\]
\[ = \left( 2 . 5p \right)^2 - \left( 1 . 5p \right)^2 + \left( 1 . 5q \right)^2 - \left( 2 . 5q \right)^2 \]
\[ = \left[ \left( 2 . 5p + 1 . 5p \right)\left( 2 . 5p - 1 . 5p \right) \right] + \left[ \left( 1 . 5q + 2 . 5 \right)\left( 1 . 5q - 2 . 5q \right) \right] \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right] \]
\[ = 4p \times p + 4q \times \left( - q \right)\]
\[ = 4 p^2 - 4 q^2 \]
\[ = 4\left( p^2 - 4 q^2 \right) \]
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