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AI-তৈরি নমুনা উত্তর

Given that $\alpha$ and $\beta$ are distinct and satisfy:
$\alpha^2 = 5\alpha - 3 \implies \alpha^2 - 5\alpha + 3 = 0$
$\beta^2 = 5\beta - 3 \implies \beta^2 - 5\beta + 3 = 0$

Thus, $\alpha$ and $\beta$ are the roots of the quadratic equation:
$x^2 - 5x + 3 = 0$

By Vieta's formulas:
$\alpha + \beta = 5$
$\alpha\beta = 3$

We need to find the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

Sum of the roots:
$S = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} = \frac{5^2 - 2(3)}{3} = \frac{25 - 6}{3} = \frac{19}{3}$

Product of the roots:
$P = \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1$

Therefore, the required quadratic equation is:
$x^2 - Sx + P = 0$
$x^2 - \frac{19}{3}x + 1 = 0$
$\implies 3x^2 - 19x + 3 = 0$