Simplify the following (x + 2y + 3) (x + 2y + 7)

by

(i) (x + 2y + 3) (x + 2y + 7)

(ii) (2x + y + 5) (2x + y – 9)

(iii) (x – 2y – 5) (x – 2y + 3)

(iv) (3x – 4y – 2) (3x – 4y – 6)

Answer :

(i) (x + 2y + 3) (x + 2y + 7)

Consider x + 2y = a

(a + 3) (a + 7) = a2 + (3 + 7) a + 3 × 7

By further calculation

= a2 + 10a + 21

Substituting the value of a

= (x + 2y)2 + 10 (x + 2y) + 21

By expanding using formula

= x2 + 4y2 + 2 × x × 2y + 10x + 20y + 21

So we get

= x2 + 4y2 + 4xy + 10x + 20y + 21

(ii) (2x + y + 5) (2x + y – 9)

Consider 2x + y = a

(a + 5) (a – 9) = a2 + (5 – 9) a + 5 × (-9)

By further calculation

= a2 – 4a – 45

Substituting the value of a

= (2x + y)2 – 4 (2x + y) – 45

By expanding using formula

= 4x2 + y2 + 2 × 2x × y – 8x – 4y – 45

So we get

= 4x2 + y2 + 4xy – 8x – 4y – 45

(iii) (x – 2y – 5) (x – 2y + 3)

Consider x – 2y = a

(a – 5) (a + 3) = a2 + (- 5 + 3) a + (-5) (3)

By further calculation

= a2 – 2a – 15

Substituting the value of a

= (x – 2y)2 – 2 (x – 2y) – 15

By expanding using formula

= x2 + 4y2 – 2 × x × 2y – 2x + 4y – 15

So we get

= x2 + 4y2 – 4xy – 2x + 4y – 15

(iv) (3x – 4y – 2) (3x – 4y – 6)

Consider 3x – 4y = a

(a – 2) (a – 6) = a2 (- 2 – 6)a + (-2) (-6)

By further calculation

= a2 – 8a + 12

Substituting the value of a

= (3x – 4y)2 – 8 (3x – 4y) + 12

Expanding using formula

= 9x2 + 16y2 – 2 × 3x × 4y – 24x + 32y + 12

So we get

= 9x2 + 16y2 – 24xy – 24x + 32y + 12

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