First slide

Theory of equations

Question

If the roots of equation $1 (a - 1) {(x^{2} + x + 1)}^{2} = (a + 1) (x^{4} + x^{2} + 1)$ are real and distinct, then the value of $a \in$

Moderate

A

$(- \infty, 3]$
B

$(- \infty, - 2) \cup (2, \infty)$
C

[-2, 2]
D

$[- 3, \infty)$

Solution

$\begin{array}{l} x^{4} + x^{2} + 1 = {(x^{2} + 1)}^{2} - x^{2} \\ = (x^{2} + x + 1) (x^{2} - x + 1) \\ x^{2} + x + 1 = {(x + \frac{1}{2})}^{2} + \frac{3}{4} \neq 0 \forall x \end{array}$
Therefore, we can cancel this factor and we get
$\begin{array}{l} (a - 1) (x^{2} - x + 1) = (a + 1) (x^{2} - x + 1) \\ or x^{2} - ax + 1 = 0 \end{array}$
It has real and distinct roots if $D = a^{2} - 4 > 0$

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