Exercise 52 Page 249

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.

x ( sqrt(3)-1 ) = ( sqrt(3)+2 ) sqrt(3)

DivEqn

.LHS /( sqrt(3)-1 ).=.RHS /( sqrt(3)-1 ).

x = ( sqrt(3)+2 ) sqrt(3)/sqrt(3)-1

▼

Distribute sqrt(3)

Distr

Distribute sqrt(3)

x = ( sqrt(3))^2 + 2sqrt(3)/sqrt(3)-1

PowSqrt

( sqrt(a) )^2 = a

x = 3 + 2sqrt(3)/sqrt(3)-1

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)

▼

Simplify denominator

(a-b)(a+b)=a^2-b^2

x = ( 3+2sqrt(3) )( sqrt(3)+1 )/(sqrt(3))^2-1^2

CalcPow

Calculate power

x = ( 3+2sqrt(3) )( sqrt(3)+1 )/3-1

SubTerm

Subtract term

x = ( 3+2sqrt(3) )( sqrt(3)+1 )/2

▼

Simplify numerator

Distr

Distribute ( sqrt(3)+1 )

x = 3( sqrt(3)+1 )+2sqrt(3)( sqrt(3)+1 )/2

Distr

Distribute 3

x = 3sqrt(3)+3+2sqrt(3)( sqrt(3)+1 )/2

Distr

Distribute 2sqrt(3)

x = 3sqrt(3)+3+2(sqrt(3))^2+2sqrt(3)/2

PowSqrt

( sqrt(a) )^2 = a

x = 3sqrt(3)+3+2(3)+2sqrt(3)/2

Multiply

Multiply

x = 3sqrt(3)+3+6+2sqrt(3)/2

AddTerms

Add terms

x = 9+5sqrt(3)/2

The solution of our equation is x = 9+5sqrt(3)2.