Maximize an Almost-Factorable Expression

Algebra Level 5

Real numbers a,b,c,da,b,c,d satisfy

a2+b2=c2+d2.a^2+b^2=c^2+d^2.

If the maximum of the expression

20bc+14bd+20ad14aca2+b2+c2+d2\dfrac{20bc+14bd+20ad-14ac}{a^2+b^2+c^2+d^2}

can be expressed as n\sqrt{n} for some positive integer nn, what is the value of n?n?

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