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
$\frac{4}{y + 1} + \frac{5}{y - 5} = - \frac{3}{2} \Rightarrow y = - 5, 3$
$x + \frac{3}{x} = - 5$ has two irrational roots and $x + \frac{3}{x} = 3$ has imaginary roots.

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