First slide

Theory of equations

Question

The number of irrational roots of the equation $\frac{4 x}{x^{2} + x + 3} + \frac{5 x}{x^{2} - 5 x + 3} = - \frac{3}{2}$ is

Moderate

A

4
B

0
C

1
D

2

Solution

Here, x = 0 is not a root. Divide both the numerator and
denominator by x and put x + 3/x = y to obtain
$\frac{4}{y + 1} + \frac{5}{y - 5} = - \frac{3}{2} \Rightarrow y = - 5, 3$
$\frac{x + 3}{x} = - 5$ has two irrational roots and $\frac{x + 3}{x} = 3$ has imaginary roots.

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