First slide

Theory of expressions

Question

lf $α$ and $β$ the roots of the equation $x^{2} + px - 1 / (2 p^{2}) = 0$ , where $p \in R$ .Then the minimum value of $α^{4} + β^{4}$ is

Moderate

A

$2 \sqrt{2}$
B

$2 - \sqrt{2}$
C

2
D

$2 + \sqrt{2}$

Solution

Here,
$\begin{array}{l} α^{4} + β^{4} = {(α^{2} + β^{2})}^{2} - 2 α^{2} β^{2} \\ = {\{(α + β)^{2} - 2 αβ\}}^{2} - 2 (αβ)^{2} \\ = {(p^{2} + \frac{1}{p^{2}})}^{2} - \frac{1}{2 p^{4}} \\ = p^{4} + \frac{1}{2 p^{4}} + 2 \\ = {(p^{2} - \frac{1}{\sqrt{2} p^{2}})}^{2} + 2 + \sqrt{2} \geq 2 + \sqrt{2} \end{array}$
Thus, the minimum value of $α^{4} + β^{4} is 2 + \sqrt{2}$

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