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

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To find (a^2+2b^2)/ab

Solution:

If a/b = x, the a = bx.

Then (a^2+2b^2b^2)/ab =[ (bx)^2+2b^2]/(bx)b

=b^2(x^2+2)/(bx)b

=b^2(x^2+2)/x

=b^2{(2sqrt3)^2+2)/(2sqrt3)

=b^2{12+2}/(2sqrt3)

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

=b^2(14)sqrt3/6

=b^2[7*3^(1/2)]/(3).

However, if you write, the the given expression,(a^2 + 2b^2)/ab, like (a^2 + 2b^2)/**(ab)**, then

(a^2+2b^2)/(ab) =[(bx)^2+2b^2]/(bx*b)

=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)

= (3^1/2)*(7/3)

= 7/(3^1/2)

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**