First slide

Theory of equations

Question

If $x^{2} + ax - 3 x - (a + 2) = 0$ has real and distinct roots, then the minimum value of $\frac{(a^{2} + 1)}{(a^{2} + 2)}$ is

Moderate

A

1
B

0
C

1/2
D

1/4

Solution

D > 0
$\Rightarrow (a - 3)^{2} + 4 (a + 2) > 0$
or $a^{2} - 6 a + 9 + 4 a + 8 > 0$
or $a^{2} - 2 a + 17 > 0$
$\begin{array}{l} \Rightarrow a \in R \\ ∴ \frac{a^{2} + 1}{a^{2} + 2} = 1 - \frac{1}{a^{2} + 2} \geq \frac{1}{2} \end{array}$

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