Chủ đề này có 1 trả lời

[quan1234]

Giải phương trình

$\sqrt{2x^2-2x+1}+\sqrt{2x^2-(\sqrt{3}-1)x+1}+\sqrt{2x^2+(\sqrt{3}+1)x+1}=3$

[VuHongQuan]

$LHS=\sqrt{x^2+(x-1)^2}+\sqrt{(\frac{\sqrt{3}}{2}-x)^2+(x+\frac{1}{2})^2}+\sqrt{(x+\frac{\sqrt{3}}{2})^2+(x+\frac{1}{2})^2}\\\ge\sqrt{x^2+(x-1)^2}+\sqrt{{(\frac{\sqrt{3}}{2}-x}+x+\frac{\sqrt{3}}{2})^2+(\frac{1}{2}+x+x+\frac{1}{2})^2}\\=\sqrt{x^2+(x-1)^2}+\sqrt{(2x+1)^2+3}\\=\sqrt{x^2+(x-1)^2}+\sqrt{3x^2+(x+2)^2}\ge\begin{vmatrix} x-1 \end{vmatrix}+\begin{vmatrix} x+2 \end{vmatrix}\ge\begin{vmatrix} 1-x+x+2 \end{vmatrix}=3=RHS$

dấu "=" xảy ra khi x=0