MULTIPLYING TWO BINOMIALS WITH RADICALS

Expand and simplify :

Problem 1 :

(1 + √2) (2 + √2)

Solution :

(1   + √2) (2 + √2)

= (1) (2 + √2) + (√2) (2 + √2)

= 2 + √2 + 2√2 + √2 ⋅ √2

= 2 + 3√2 + 2

= 4 + 3√2

Problem 2 :

(2 + √3) (3 + √3)

Solution :

(2 + √3) (3 + √3)

= 2 (3 + √3) + (√3) (3 + √3)

= 6 + 2√3 + 3√3 + √3 ⋅√3

= 6 + 5√3 + 3

= 9 + 5√3

Problem 3 :

(4 - √2) (3 + √2)

Solution :

(4 - √2) (3 + √2)

= 4(3 + √2) - √2(3 + √2)

= 12 + 4√2 - 3√2 - √2 ⋅√2

= 12 + √2 – 2

= 10 + √2 

Problem 4 :

(1 + √3) (1 - √3)

Solution :

(1 + √3) (1 - √3)

= (1)(1 - √3) + (√3)(1 - √3)

= 1 - √3 + √3 - √3 ⋅√3

= 1 - 3

= -2

Problem 5 :

(√5 + 2) (√5 - 3)

Solution :

(√5 + 2) (√5 - 3)

= (√5)(√5 - 3) + 2 (√5 - 3)

= √5 ⋅√5 - 3√5 + 2√5 – 6

= 5 - √5 – 6

= -1 - √5

Problem 6 :

(6 - √3) (2 + √3)

Solution :

(6 - √3) (2 + √3)

= 6(2 + √3) - √3(2 + √3)

= 12 + 6√3 - 2√3 - √3 ⋅√3

= 12 + 4√3 – 3

= 9 + 4√3 

Problem 7 :

(4 - 3√3) (2 - √3)

Solution :

(4 - 3√3) (2 - √3)

= (4) (2 - √3) - (3√3) (2 - √3)

= 8 - 4√3 - 6√3 + 3(√3 ⋅ √3)

= 8 - 10√3 + 9

= 17 - 10√3

Problem 8 :

(-1 + 2√2) (2 - √2)

Solution :

(-1 + 2√2) (2 - √2)

= (-1)(2 - √2) + 2√2(2 - √2)

= -2 + √2 + 4√2 – 2(√2 ⋅ √2)

= -2 + 5√2 – 4

= -6 + 5√2