First slide

Theory of equations

Question

The number of rational roots of
$81 {(\frac{2 x - 5}{3 x + 1})}^{4} - 45 {(\frac{2 x - 5}{3 x + 1})}^{2} + 4 = 0, x \neq 1 / 3$ is

Moderate

A

1
B

2
C

3
D

4

Solution

Put ${(\frac{2 x - 5}{3 x + 1})}^{2} = t$ so that equation becomes
$\begin{array}{l} 81 t^{2} - 45 t + 4 = 0 \\ \Rightarrow 81 t^{2} - 36 t - 9 t + 4 = 0 \end{array}$
$\Rightarrow t = \frac{1}{9}, \frac{4}{9} \Rightarrow \frac{2 x - 2}{3 x + 1} = \pm \frac{1}{3}, \pm \frac{2}{3}$
Thus, the given equation has four rational roots

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