soit ` x ` et ` y ` deux réels tels que `x = sqrt(3) -1 ` et `y = 1/(1-sqrt(5))`

Déterminer `abs(x)` , `abs(y)` , `abs(x+y )` , `abs(x-y)` , `abs(xy)` , `abs(x/y)`

Déterminer ` abs(x) `

Méthode

Pour comparer deux réels positifs `a` et `b` il suffit de comparer leur carrés

c 'est à dire pour tous réels `a` et ` b` positifs : ` a^2 >= b^2 ` équivalent à ` a >= b `



On a ` x = sqrt(3) -1 `

comme ` sqrt(3)^2 = 3 ` et `1^2 = 1 `

donc ` sqrt(3)^2 > 1^2 `

or ` sqrt(3) ` et ` 1 ` sont positifs

alors ` sqrt(3) > 1 `

` sqrt(3) -1 > 0 `

` abs(sqrt(3) -1) = sqrt(3) -1 `

` abs(x)= sqrt(3) -1 `

Déterminer ` abs(y) `

Rappels

1 le conjuguée de ` sqrt(a) +sqrt(b)` est ` sqrt(a)-sqrt(b)`

2 le conjuguée de ` sqrt(a) - sqrt(b)` est ` sqrt(a) +sqrt(b)`



On a ` y =1/(1 -sqrt(5)) `

` = (1+sqrt(5))/((1 -sqrt(5))(1 +sqrt(5)))`

` = (1+sqrt(5))/(1^1 -sqrt(5)^2) ` `(a-b)(a+b)= a^2 -b^2 `

` = (1+sqrt(5))/(1 -5 )`

` = (1+sqrt(5))/(-4) `

` = - [(1+sqrt(5))/4]`

Comme ` [(1+sqrt(5))/4] > 0 `

alors ` - [(1+sqrt(5))/4] < 0 `

il s'ensuit que ` y < 0 `

`=> abs(y)= - y = - ( - [(1+sqrt(5))/4]) = [(1+sqrt(5))/4]`

`=> abs(y)= (1+sqrt(5))/4`

Déterminer `abs(x+y) `

On a ` x = sqrt(3) -1 ` et ` y = 1/(1-sqrt(5)) = - (1+sqrt(5))/4 `

` x +y = sqrt(3) -1 - (1+sqrt(5))/4 `

` = (4(sqrt(3) -1) - (1+sqrt(5)) )/4 `

` = (4sqrt(3) -4 - (1+sqrt(5)))/4 `

` = (4sqrt(3) -4 - 1- sqrt(5))/4 `

` = (4sqrt(3) - 5 -sqrt(5))/4 `

` = (4sqrt(3) -(5+sqrt(5)))/4 `

Etudions le signe de `4sqrt(3) - (5+sqrt(5))`

On a `(4sqrt(3))^2 = 4^2 xxsqrt(3)^2 = 16 xx3 = 48 `

on a ` (5 +sqrt(5))^2 = 5^2 +2xx5xxsqrt(5) + sqrt(5)^2 = 25+5 +10sqrt(5) = 30 +10sqrt(5) `

on a ` 48 - (30+10sqrt(5)) = 18 -10sqrt(5) `

comme `18^2 = 324` et ` (10sqrt(5))^2 = 10^2xxsqrt(5)^2 = 100xx 5 = 500 `

alors ` (10sqrt(5))^2 > 18^2 `

`=> 10sqrt(5) > 18 `

`=> 18 -10sqrt(5) < 0 `

`=> 48 - (30+10sqrt(5)) < 0 `

`=> (4sqrt(3))^2 - (5 +sqrt(5))^2 < 0 `

`=> (4sqrt(3))^2 < (5 +sqrt(5))^2 `

`=> 4sqrt(3) < 5+sqrt(5)`

` => (4sqrt(3) -(5+sqrt(5)))/4 < 0 `

`=> x +y < 0 `

et par suite ` abs(x+y)= -(x+y) = - (4sqrt(3) -(5+sqrt(5)))/4 = ( 5+sqrt(5) -4sqrt(3))/4 `

`=> abs(x+y)= ( 5+sqrt(5) -4sqrt(3))/4 `