Question

If `x=2+sqrt(3)` and `y=2-sqrt(3)` then what is `x^2+y^2+xy` ?

Answer 1

15

Explanation:

`x^2+y^2+xy= (x+y)^2-xy`

`= (2+sqrt(3)+2-sqrt3)^2-[(2+sqrt3)(2-sqrt3)]`

`=16-(4-3)=15`

used: `(a+b)^2=a^2+b^2+2ab`
`(a+b)(a-b)=a^2-b^2`

Answer 2

`15`

Explanation:

If `x=2+sqrt(3)`
`color(white)("XXXXXXXX")x^2=4+4sqrt(3)+3=color(red)( 7+4sqrt(3))`

If `y=2-sqrt(3)`
`color(white)("XXXXXXXX")y^2=4-4sqrt(3)+3= color(red)(7 - 4sqrt(3))`

and
`color(white)("XXXXXXXX")xy = 4-3 color(white)("XXXXX")=color(red)(1)`

Sum`=color(white)("XXXXXXXXXXXXXXXX")= color(blue)(15)`

Answer 3

`x^2+y^2+xy = 15`

Explanation:

Use the difference of squares identity:

`a^2 - b^2 = (a+b)(a-b)` to calculate `xy`:

`xy = (2+sqrt(3))(2-sqrt(3)) = 2^2-sqrt(3)^2 = 4-3 = 1`

`color(white)()`
Also `(x+y)^2 = x^2+2xy+y^2`

So:

`x^2+y^2+xy = x^2+2xy+y^2 - xy`

`=(x+y)^2 - xy = 4^2-1 = 16-1 = 15`