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

3
B

0
C

1
D

2

Solution

$\frac{4 x}{x (x + 1 + \frac{3}{x})} + \frac{5 x}{x (x - 5 + \frac{3}{x})} = - \frac{3}{2}$
x = 0 is not a root. Divide both the numerators and
denominators by x and put x + $\frac{3}{x}$ = y to obtain
$\frac{4}{y + 1} + \frac{5}{y - 5} = - \frac{3}{2} \Rightarrow y = - 5, 3$
x + $\frac{3}{x}$ = - 5
$\frac{x^{2} + 3}{x} = - 5 x^{2} + 5 x + 3 = 0 x = \frac{- 5 \pm \sqrt{25 - 12}}{2} = \frac{- 5 \pm \sqrt{13}}{2}$
has two irrational roots and
x + $\frac{3}{x}$ = 3
$\frac{x^{2} + 3}{x} = 3 x^{2} - 3 x + 3 = 0 x = \frac{3 \pm \sqrt{9 - 12}}{2} = \frac{3 \pm \sqrt{- 3}}{2} = \frac{3 \pm i \sqrt{3}}{2}$
has imaginary roots.

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