To find (a^2+2b^2)/ab
If a/b = x, the a = bx.
Then (a^2+2b^2b^2)/ab =[ (bx)^2+2b^2]/(bx)b
However, if you write, the the given expression,(a^2 + 2b^2)/ab, like (a^2 + 2b^2)/(ab), then
=b^2(x^2+2)/(b^2 *x) =(x^2+2)/x = 7(sqrt3)/3
We have to find value of expression E = (a^2 + 2b^2)/ab
given a/b = 2*3^1/2.
We firsts simplify the given expression as indicated in following steps.
E = (a^2 + 2b^2)/ab
= (a^2)/ab + (2b^2)/ab
= a/b + 2/(a/b)
Substituting given value of a/b in this expression we get:
E = 2*3^1/2 + 2/(2*3^1/2)
= 2*3^1/2 + 1/(3^1/2)
= (3^1/2)*(2 + 1/3)
For calculating the expression, we’ll choose to express “a”depending on “b”,
So, from the given facts of the problem, we know that:
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”.
After reducing the similar terms:
We’ll amplify with sqrt3, in order not to keep a "sqrt" to the denominator, which is not indicated.