What is the value for the ratio (a^2 + 2b^2)/ab if the value of the ratio a/b = 2sqrt3

For calculating the expression, we’ll choose to express “a”depending on “b”,

So, from the given facts of the problem, we know that:

a/b=2*sqrt3

We’ll cross multiply and  we’ll have:

a=2*b*sqrt3 => a^2 =4*b^2*3 => a^2 =12*b^2

Now we’ll substitute a and a^2 in the expression which has to be calculated:

E(a,b)=(a^2 + 2*b^2)/a*b= (12*b^2 + 2*b^2)/2*b^2*sqrt3

We’ve noticed that the expression in a and b, have been transformed into an expression depending only by “b”.

E(b)=14*b^2/2*b^2*sqrt3

 After reducing the similar terms:

E(b)=7/sqrt3

We’ll amplify  with sqrt3, in order not to keep a "sqrt" to the denominator, which is not indicated.

E(b)=7*sqrt3/sqrt3*sqrt3

E(b)=7*sqrt3/3