If $$\alpha, \beta$$ are the roots of the equation $$x^2 - (5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}})x + 3(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1) = 0$$ then the equation, whose roots are $$\alpha + \dfrac{1}{\beta}$$ and $$\beta + \dfrac{1}{\alpha}$$, is

Solution :

Given quadratic equation :

$$x^2-\left(5+3^{\sqrt{\log_3 5}}-5^{\sqrt{\log_5 3}}\right)x + 3\left(3^{(\log_3 5)^{1/3}}-5^{(\log_5 3)^{2/3}}-1\right)=0$$

First simplify the coefficients.

Using identity :

$$a^{\log_a b}=b$$

Also,

$$3^{\sqrt{\log_3 5}} = 5^{\sqrt{\log_5 3}}$$

Therefore,

coefficient of $$x$$ becomes :

$$5+3^{\sqrt{\log_3 5}}-5^{\sqrt{\log_5 3}}=5$$

Hence,

$$\alpha+\beta = 5$$

Now consider constant term :

$$3^{(\log_3 5)^{1/3}} = 5^{(\log_5 3)^{2/3}}$$

Thus,

$$3\left(3^{(\log_3 5)^{1/3}}-5^{(\log_5 3)^{2/3}}-1\right)$$

$$=3(0-1)$$

$$=-3$$

Therefore,

$$\alpha\beta=-3$$

Now required roots are :

$$\alpha+\frac1\beta$$

and

$$\beta+\frac1\alpha$$

Sum of roots :$$S = \alpha+\beta+\frac1\alpha+\frac1\beta$$

$$= (\alpha+\beta)+\frac{\alpha+\beta}{\alpha\beta}$$

$$= 5+\frac{5}{-3}$$

$$= \frac{10}{3}$$

Product of roots :

$$P = \left(\alpha+\frac1\beta\right) \left(\beta+\frac1\alpha\right)$$

$$=\alpha\beta+1+1+\frac1{\alpha\beta}$$

$$=-3+2-\frac13$$

$$= -\frac43$$

Hence required equation :

$$x^2-Sx+P=0$$

$$x^2-\frac{10}{3}x-\frac43=0$$

Multiplying by 3 :

$$3x^2-10x-4=0$$

Final Answer :

$$3x^2-10x-4=0$$