First slide

Theory of equations

Question

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

Moderate

A

$(- \infty, 3]$
B

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

$[- 2, 2]$
D

$[- 3, \infty)$

Solution

$\begin{array}{l} x^{4} + x^{2} + 1 = {(x^{2} + 1)}^{2} - x^{2} = (x^{2} + x + 1) (x^{2} - x + 1) \\ x^{2} + x + 1 = {(x + \frac{1}{2})}^{2} + \frac{3}{4} \neq 0 \forall real x \end{array}$
Therefore we can cancel this factor and we get
$(a - 1) (x^{2} + x + 1) = (a + 1) (x^{2} - x + 1)$
$a (x^{2} + x + 1 - x^{2} + x - 1) = x^{2} - x + 1 + x^{2} + x + 1$
$a (2 x) = 2 x^{2} + 2$
$x^{2} - a x + 1 = 0$
$\begin{aligned} x^{2} - a x + 1 = 0 has real and distinct roots \\ \Rightarrow D = a^{2} - 4 > 0 \Rightarrow a \in (- \infty, - 20) \cup (2, \infty) \end{aligned}$

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