First slide

Theory of equations

Question

$If α \neq β and α^{2} = 5 α - 3, β^{2} = 5 β - 3, then$
the equation whose roots are $\frac{α}{β} and \frac{β}{α}$ is

Moderate

A

$3 x^{2} - 25 x + 3 = 0$
B

$x^{2} - 5 x + 3 = 0$
C

$x^{2} + 5 x - 3 = 0$
D

$3 x^{2} - 19 x + 3 = 0$

Solution

$α, β are roots of x^{2} = 5 x - 3 or x^{2} - 5 x + 3 =$ 0. Thus,
$α + β = 5, α β = 3$
$\begin{array}{l} Next, \frac{α}{β} + \frac{β}{α} = \frac{α^{2} + β^{2}}{α β} = \frac{(α + β)^{2}}{α β} - 2 \\ = \frac{25}{3} - 2 = \frac{19}{3} \end{array}$
and $(\frac{α}{β}) (\frac{β}{α}) = 1$
Thus, the quadratic equation whose roots are $\frac{α}{β}, \frac{β}{α}$ is
$x^{2} - \frac{19}{3} x + 1 = 0 or 3 x^{2} - 19 x + 3 = 0$

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