a) Không giải phương trình, hãy xét dấu hiệu của phương trình sau (nếu có): \(
(2-\sqrt{3})x^{2}+2(\sqrt{3}-1)x+1=0
\).
b) Không giải phương trình, hãy cho biết phương trình sau có bao nhiêu nghiệm?
             \(
\sqrt{3}x^{4}+3(\sqrt{2}-\sqrt{5})x^{2}-\sqrt{11}=0
\) (1).
                                         Giải
a)  Ta có: \(
\Delta'=2-\sqrt{3}>0;P=\frac{1}{2-\sqrt{3}}>0;S=-\frac{2(\sqrt{3}-1)}{2-\sqrt{3}}<0
\).
Vậy phương trình có hai nghiệm âm phân biệt.
b) Đặt \(
t=x^{2},t\geq 0
\). Phương trình (1) trở thành:
          \(
\sqrt{3}t^{2}+3(\sqrt{2}-\sqrt{5})t-\sqrt{11}=0  
\)   (2)
  \(
P=\frac{c}{a}=-\frac{\sqrt{11}}{\sqrt{3}}<0\Rightarrow
\) phương trình (2) có hai  nghiệm  \(
t_{1}<0<t_{2}
\)
\(
\Rightarrow
\) nhận \(t_{2}>0 \Rightarrow \) phương trình (1) có hai nghiệm phân biệt. 

Thẻ

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