Question

Find the following product and verify the result for x = − 1, y = − 2:
(3x − 5y) (x + y)

Answer 1

To multiply, we will use distributive law as follows:

\[\left( 3x - 5y \right)\left( x + y \right)\]

\[ = 3x\left( x + y \right) - 5y\left( x + y \right)\]

\[ = 3 x^2 + 3xy - 5xy - 5 y^2 \]

\[ = 3 x^2 - 2xy - 5 y^2\] 

\[\therefore\] \[\left( 3x - 5y \right)\left( x + y \right) = 3 x^2 - 2xy - 5 y^2\].

Now, we put x = \[-\] 1 and y = \[-\] 2 on both sides to verify the result.

\[\text { LHS } = \left( 3x - 5y \right)\left( x + y \right)\]

\[ = \left\{ 3\left( - 1 \right) - 5\left( - 2 \right) \right\}\left\{ - 1 + \left( - 2 \right) \right\}\]

\[ = \left( - 3 + 10 \right)\left( - 3 \right)\]

\[ = \left( 7 \right)\left( - 3 \right)\]

\[ = - 21\]

\[\text { RHS } = 3 x^2 - 2xy - 5 y^2 \]

\[ = 3 \left( - 1 \right)^2 - 2\left( - 1 \right)\left( - 2 \right) - 5 \left( - 2 \right)^2 \]

\[ = 3 \times 1 - 4 - 5 \times 4\]

\[ = 3 - 4 - 20\]

\[ = - 21\]

Because LHS is equal to RHS, the result is verified.
Thus, the answer is \[3 x^2 - 2xy - 5 y^2\].