First slide

Theory of equations

Question

$If the absolute value of the difference of roots of the equation x^{2} + p x + 1 = 0 exceeds \sqrt{3 p} then$

Moderate

A

$- 1 < p < 4$
B

$0 \leq p < 4$
C

$p < - 1 or p > 4$
D

$p > 4$

Solution

$\begin{array}{l} | α - β | > \sqrt{3 p} \\ \Rightarrow (α + β)^{2} - 4 α β > 3 p \\ \Rightarrow p^{2} - 3 p - 4 > 0 and p > 0 \\ \Rightarrow (p - 4) (p + 1) > 0 and p > 0 \\ \Rightarrow p > 4 \end{array}$

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