\(\text{Δ}=\left(2\sqrt{3}-2\right)^2-4\sqrt{3}\cdot\left(-4\right)\)
\(=12-8\sqrt{3}+4+16\sqrt{3}\)
\(=12+8\sqrt{3}+4\)
\(=\left(2\sqrt{3}+2\right)^2\)
Vì Δ>0 nên phương trình có hai nghiệm phân biệt là:
\(\left\{{}\begin{matrix}x_1=\dfrac{-2\left(\sqrt{3}-1\right)-2\sqrt{3}-2}{2\sqrt{3}}=\dfrac{-4\sqrt{3}}{2\sqrt{3}}=-2\\x_2=\dfrac{-2\left(\sqrt{3}-1\right)+2\sqrt{3}+2}{2\sqrt{3}}=\dfrac{-2\sqrt{3}+2+2\sqrt{3}+2}{2\sqrt{3}}=\dfrac{4}{2\sqrt{3}}=\dfrac{2\sqrt{3}}{3}\end{matrix}\right.\)
\(\sqrt{3}\)x\(^2\)+2\((\)\(\sqrt{3}-1\)\()x\)-4=0
cos \(\Delta\)'=b'\(^2\)-ac=\((\sqrt{3}-1)^2-\sqrt{3}\cdot(-4)\)
=3-2\(\sqrt{3}\)+1+4\(\sqrt{3}\)
=2\(\sqrt{3}\) +4
=3+2\(\sqrt{3}\) +1
=\((\sqrt{3}+1)^2\)\(\ge0\)
\(\Rightarrow\)Phuong trinh co â nghiem phan biet
x1=-\(\sqrt{3}+1\)+\(\sqrt{3}+1\)/\(\sqrt{3}\)
=2\(\sqrt{3}\)/3
x2=-\(\sqrt{3}\)+1-\(\sqrt{3}-1\)/\(\sqrt{3}\)
=-2\(\sqrt{3}\)/\(\sqrt{3}\)
=-2
vay x1=2\(\sqrt{3}\)/3;x2=-2 la nghiem cua phuong trinh
