First slide

Theory of equations

Question

The number irrational roots of
${(x^{2} + 3 x + 2)}^{2} - 8 (x^{2} + 3 x) - 4 = 0$ is

Moderate

A

0
B

2
C

3
D

4

Solution

Put $x^{2} + 3 x + 2 = t$ to obtain $t^{2} - 8 (t - 2) - 4 = 0$
$\begin{array}{l} \Rightarrow t^{2} - 8 t + 12 = 0 \Rightarrow (t - 2) (t - 6) = 0 \\ \Rightarrow t = 2, 6 \end{array}$
But both $x^{2} + 3 x + 2 = 2$
and $x^{2} + 3 x + 2 = 6$ have integral roots.

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