Question

Cho x² + y² + z² = 10. Tính:

P = ( xy + yz + xz)² + ( x - yz)² + ( y - xz)² + ( z - xy)²

Answer 1

Giải:

Sửa đề:

\(P=\left(xy+yz+xz\right)^2+\left(x^2-yz\right)^2+\left(y^2-xz\right)^2+\left(z^2-xy\right)^2\)

\(\Leftrightarrow P=x^2y^2+y^2z^2+x^2z^2+2xy^2z+2x^2yz+2xyz^2+x^4-2x^2yz+y^2z^2+y^4-2xzy^2+x^2z^2+z^4-2xyz^2+x^2y^2\)

\(\Leftrightarrow P=2x^2y^2+2y^2z^2+2x^2z^2+x^4+y^4+z^4\)

\(\Leftrightarrow P=\left(x^2+y^2+z^2\right)^2\)

\(\Leftrightarrow P=10^2\)

\(\Leftrightarrow P=100\)

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