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Understand the problem

.Find all real numbers x for which
$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$

Source of the problem

C.M.I (Chennai mathematical institute ) U.G-2019

Algebra

8 out of 10

Suggested Book

.

Do you really need a hint? Try it first!

It is of the the form of

$\frac{a^3+b^3}{a^2b+b^2a}$ .

Do you observe ?

where a=$$2^x$$

b=$$3^x$$

$$\frac{a^3+b^3+a^2b+b^2a}{a^2b+b^2a}$$=$$\frac{13}{6}$$

$$\Rightarrow\frac{a^2+b^2}{ab}=\frac{13}{6}$$

$$\frac{a}{b}+\frac{b}{a}=\frac{13}{6}$$

let x=a/b then it is a quadratic equation

$$6x^2-13x+6=0$$

on solving we get x=3/2 or 2/3

now replace x by a/b

so again putting value of a and b we get final ans is +1 and -1

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