giải pt sau : \(x^2+\sqrt[]{x-2}=x+2\sqrt[]{x^2-x-1}\)
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\(C=\sqrt{4-2\sqrt{3}}-\sqrt{7+4\sqrt{3}}\)
\(\Leftrightarrow C=\sqrt{3-2\sqrt{3}+1}-\sqrt{4+4\sqrt{3}+3}\)
\(\Leftrightarrow C=\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{\left(2+\sqrt{3}\right)^2}\)
\(\Leftrightarrow C=\left|\sqrt{3}-1\right|-\left|2+\sqrt{3}\right|\)
\(\Leftrightarrow C=\sqrt{3}-1-2-\sqrt{3}\)
\(\Leftrightarrow C=-3\)
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áp dụng bđt Min-cốp-xki ta có \(\sqrt{x^2+xy+y^2}+\sqrt{x^2+xz+z^2}=\sqrt{\left(x^2+xy+\frac{y^2}{4}\right)+\frac{3y^2}{4}}+\sqrt{\left(x^2+xz+\frac{z^2}{4}\right)+\frac{3z^2}{4}}\)\(=\sqrt{\left(x+\frac{y}{2}\right)^2+\left(\frac{\sqrt{3}y}{2}\right)^2}+\sqrt{\left(-x-\frac{z}{2}\right)^2+\left(\frac{\sqrt{3}z}{2}\right)^2}\)\(\ge\sqrt{\left(x+\frac{y}{2}-x-\frac{z}{2}\right)^2+\left(\frac{\sqrt{3}y}{2}+\frac{\sqrt{3}z}{2}\right)^2}=\sqrt{\frac{y^2}{4}-\frac{yz}{2}+\frac{z^2}{4}+\frac{3y^2}{4}+\frac{3yz}{2}+\frac{3z^2}{4}}\)
\(=\sqrt{y^2+yz+z^2}\)