We are asked to explain how to solve the given equation.
sqrt(3) + 2/x = sqrt(3)-1/sqrt(3)
To solve any equation, we want to isolate the variable on one side. In the given equation, the variable x is in the denominator. Therefore, we will rewrite the equation using cross products.
sqrt(3)+2/x = sqrt(3)-1/sqrt(3)
⇓
x ( sqrt(3)-1) = ( sqrt(3)+2) sqrt(3)
Note that this is a linear equation. It is equivalent to the given equation, so it has the same solution. However, we need to remember that x cannot be equal to 0. To solve the linear equation we isolate x using the Properties of Equality.
Although we have isolated the variable x, this is not yet a valid solution. This is because it has a radical in the denominator. Therefore, we need to rationalize the denominator. When there is a binomial denominator, we expand the fraction by the conjugate of the denominator. Recall that we can find the conjugate by changing the sign of the second term of the expression.
3 + 2sqrt(3)/sqrt(3) - 1 * sqrt(3) + 1/sqrt(3) + 1
=
( 3+2sqrt(3) )( sqrt(3)+1 )/(sqrt(3)-1)( sqrt(3)+1)
Finally, to write x in the simplest form, we will evaluate the numerator and the denomianator of the obtained expression.
x = ( 3+2sqrt(3) )( sqrt(3)+1 )/(sqrt(3)-1)(sqrt(3)+1)
