First slide

Theory of equations

Question

Number of distinct real solutions of the equation $x^{2} + {(\frac{x}{x - 1})}^{2} = 8 is$

Moderate

A

1
B

2
C

3
D

4

Solution

$\begin{array}{l} x^{2} + {(\frac{x}{x - 1})}^{2} = 8 \\ \Rightarrow {(x + \frac{x}{x - 1})}^{2} - 2 x (\frac{x}{x - 1}) = 8 \\ \Rightarrow {(\frac{x^{2} - x + x}{x - 1})}^{2} - 2 (\frac{x^{2}}{x - 1}) - 8 = 0 \\ \Rightarrow {(\frac{x^{2}}{x - 1})}^{2} - 2 (\frac{x^{2}}{x - 1}) - 8 = 0 \end{array}$
$\begin{array}{l} \frac{x^{2}}{x - 1} = t \Rightarrow t^{2} - 2 t - 8 = 0 \Rightarrow t = 4, t = - 2 \\ p u t \frac{x^{2}}{x - 1} = 4 \Rightarrow x^{2} - 4 x + 4 = 0 \Rightarrow x = 2 \end{array}$
$\begin{array}{l} Put \frac{x^{2}}{x - 1} = - 2 \Rightarrow x^{2} + 2 x = 2 \Rightarrow {(x + 1)}^{2} = 3 \\ \Rightarrow x + 1 = \pm \sqrt{3} \\ \Rightarrow x = \pm \sqrt{3} - 1 \end{array}$

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