Question

If \[\frac{5 - \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3}\] , then

Options

•  x = 13, y = −7

• x = −13, y = 7

• x = −13, y =- 7

• x = 13, y = 7

Answer 1

Given that:`(5-sqrt3)/(2+sqrt3) = x+ysqrt3`We need to find x and y

We know that rationalization factor for `2+sqrt3` is`2-sqrt3` . We will multiply numerator and denominator of the given expression `(5-sqrt3)/(2+sqrt3)`by, 2-sqrt3`  to get

`(5-sqrt3)/(2+sqrt3) xx (2-sqrt3)/(2-sqrt2)  = (5 xx 2 - 5 xx sqrt3 - 2 xx sqrt3 +(sqrt3)^3)/((2)^2 - (sqrt3)^2)`

` (10-5sqrt3 - 2 sqrt3 +3)/((2)^2 -(sqrt3)^2)`

` = (13-7sqrt3) /(4-3)`

` = 13 - 7sqrt3.`

Since ` x + y sqrt3 = 13 - 7 sqrt3`

On equating rational and irrational terms, we get  `x=13 and  y= -7`