If alpha^2 = 5alpha - 3 and beta^2 = 5beta - | vedclass

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(A) Given that $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$,it implies that $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 -

Vedclass · Accepted Answer

$19/3$

Vedclass · Answer

(A) Given that $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$,it implies that $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 5x + 3 = 0$.From the properties of quadratic equations,the sum of the roots is $\alpha + \beta = -(-5)/1 = 5$ and the product of the roots is $\alpha \beta = 3/1 = 3$.We need to find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.This expression can be simplified as $\frac{\alpha^2 + \beta^2}{\alpha \beta}$.Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$,we substitute the known values:$\alpha^2 + \beta^2 = (5)^2 - 2(3) = 25 - 6 = 19$.Therefore,$\frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{19}{3}$.

If $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ where $\alpha \neq \beta$,what is the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$?

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