USE THE FOIL TO MULTIPLY BINOMIALS WITH RADICALS

Expand and simplify :

Problem 1 :

(1 + √2) (2 + √2)

Solution :

(1 + √2) (2 + √2)

= 2 + √2 + 2√2 + (√2)²

= 2 + 3√2 + 2

= 4 + 3√2

So, the answer is 4 + 3√2.

Problem 2 :

(2 + √3) (2 + √3)

Solution :

(2 + √3) (2 + √3)

= 4 + 2√3 + 2√3 + (√3)²

= 4 + 4√3 + 3

= 7 + 4√3

So, the answer is 7 + 4√3.

Problem 3 :

(√3 + 2) (√3 - 1)

Solution :

(√3 + 2) (√3 - 1)

= (√3)² - √3 + 2√3 - 2

= 3 + √3 - 2

= 1 + √3

So, the answer is 1 + √3.

Problem 4  :

(4 - √2) (3 + √2)

Solution :

(4 - √2) (3 + √2)

= 12 + 4√2 - 3√2 - (√2)²

= 12 + √2 - 2

= 10 + √2

So, the answer is 10 + √2.

Problem 5 :

(1 + √3) (1 - √3)

Solution :

(1 + √3) (1 - √3)

= 1 - √3 + √3 - (√3)²

= 1 - 3

= -2

So, the answer is -2.

Problem 6 :

(5 + √7) (2 - √7)

Solution :

(5 + √7) (2 - √7)

= 10 - 5√7 + 2√7 - (√7)²

= 10 - 3√7 - 7

 = 3 - 3√7

So, the answer is 3 - 3√7.

Problem 7 :

(√5 + 2) (√5 - 3)

Solution :

(√5 + 2) (√5 - 3)

= (√5)² - 3√5 + 2√5 - 6

= 5 - √5 - 6

= -1 - √5

So, the answer is -1 - √5.

Problem 8 :

(√7 - √3) (√7 + √3)

Solution :

(√7 -√3) (√7 + √3)

= (√7)² + (√7) (√3) - (√3) (√7) - (√3)²

= 7 - 3

= 4

So, the answer is 4.

Problem 9 :

(2√2 + √3) (2√2 - √3)

Solution :

(2√2 + √3) (2√2 - √3)

= 4(√2)² - 2√6 + 2√6 - (√3)²

= 8 - 3

= 5

So, the answer is 5.

Problem 10  :

(4 - √2) (3 - √2)

Solution :

(4 - √2) (3 - √2)

= 12 - 4√2 - 3√2 + (√2)²

= 12 - 7√2 + 2

= 14 - 7√2

So, the answer is 14 - 7√2.