Question

Express the following below

Answer 1

p = 5 ; q = -11

Explanation:

Simplify the radicals:

`\sf (a - b)^2 = a^2 + b^2 - 2ab\\\\(√(5)-3)^2 = (√(5))^2+3^2-2*√(5)*3`

`\sf = 5 + 9 -6√(5)\\\\= 14 - 6√(5)`

`\sf ((√(5)-3)^2)/(√(5)+1)=(14-6√(5))/(√(5)+1)`

Multiply the denominator and the numerator by (√5 - 1).

`\sf= ((14-6√(5))(√(5)-1))/((√(5)+1)(√(5)-1))\\\\\\= (14*√(5)-14*1-6√(5)*√(5)+6√(5)*1)/((√(5))^2-1^2)\\\\\\=(14√(5)-14-6*5+6√(5))/(5-1)\\\\\\=(14√(5)+6√(5)-14-30)/(4)`

`\sf = ((14+6)√(5)-44)/(4)\\\\\\= (20√(5)-44)/(4)\\\\=(20√(5))/(4)-(44)/(4)\\\\\\=5√(5)-11`

p = 5

q = -11