First slide

Theory of equations

Question

If $α \neq β$ and $α^{2} = 5 α - 3, β^{2} = 5 β - 3$ , then the equation having $α / β$ and $β / α$ as its roots, is

Moderate

A

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

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

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

$x^{2} - 16 x + 1 = 0$

Solution

We have
$α^{2} = 5 α - 3 \Rightarrow α^{2} - 5 α + 3 = 0 \Rightarrow α = \frac{5 \pm \sqrt{13}}{2}$
Similarly, $β^{2} = 5 β - 3 \Rightarrow β = \frac{5 \pm \sqrt{13}}{2}$
Since, $α \neq β$
$∴ α = \frac{5 + \sqrt{13}}{2}$ and $β = \frac{5 - \sqrt{13}}{2}$
or, $α = \frac{5 - \sqrt{13}}{2}$ and $β = \frac{5 + \sqrt{13}}{2}$
Thus, in either case, we obtain
$α^{2} + β^{2} = \frac{1}{4} (50 + 26) = 19$ ,
and, $α β = \frac{1}{4} (25 - 13) = 3$ , in both the cases.
Thus, the equation having $a / β a n d β / a$ as its roots is
$x^{2} - x (\frac{α}{β} + \frac{β}{α}) + \frac{α β}{α β} = 0$
$\Rightarrow x^{2} - x (\frac{α^{2} + β^{2}}{α β}) + 1 = 0 \Rightarrow 3 x^{2} - 19 x + 3 = 0$

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