First slide

Theory of equations

Question

The number of irrational roots of the equation $(x - 1) (x - 2) (3 x - 2) (3 x + 1) = 21$ is

Moderate

A

0
B

2
C

3
D

4

Solution

$\begin{aligned} (3 x - 2) (x - 1) (3 x + 1) (x - 2) = 21 \\ \Rightarrow (3 x^{2} - 5 x + 2) (3 x^{2} - 5 x - 2) = 21 \end{aligned}$
$\begin{array}{l} \Rightarrow {(3 x^{2} - 5 x)}^{2} - 4 = 21 \Rightarrow 3 x^{2} - 5 x = \pm 5 \\ \Rightarrow 3 x^{2} - 5 x - 5 = 0 or 3 x^{2} - 5 x + 5 = 0 \end{array}$
The first equation has irrational roots and the second has imaginary roots.

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