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if x + y + z = 1 and x^2 + y^2 + z^2 = 3, then find the value of xy + xz + yz. ​

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if x + y + z = 1 and x^2 + y^2 + z^2 = 3, then find the value of xy + xz + yz.

Hint: Expand (x + y + z)^2 first and compare left and right sides of the equation.

Jun 10, 2019

#1
+25342
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if x + y + z = 1 and x^2 + y^2 + z^2 = 3,
then find the value of xy + xz + yz. ​

Expand (x + y + z)^2

$$\begin{array}{rcll} (x + y + z)^2 &=& 1 \\ x^2+y^2+z^2+2(xy+yz+xz) &=& 1 \quad | \quad x^2 + y^2 + z^2 = 3 \\ 3+2(xy+yz+xz) &=& 1 \\ 2(xy+yz+xz) &=& -2 \\ \mathbf{xy+yz+xz} &=& \mathbf{-1} \\ \end{array}$$

Jun 10, 2019