Property 7
- \(\left(a+b+c\right)^2=a^2+b^2+c^2+2\left(ab+ac+bc\right)\)
- \(\left(a+b-c\right)^2=a^2+b^2+c^2+2\left(ab-ac-bc\right)\)
- \(\left(a-b-c\right)^2=a^2+b^2+c^2+2\left(-ab-ac+bc\right)\)
1.
\(a+b-c=9\)
\(a^2+b^2+c^2=23\)
\(\Rightarrow ac+bc-ab=?\)
2.
\(a-b+c=11\)
\(a^2+b^2+c^2=15\)
\(\Rightarrow ac-bc-ab=?\)
3.
\(a-b-c=8\)
\(ab+ac-bc=-10\)
\(\Rightarrow a^2+b^2+c^2=?\)
4.
\(a^2+b^2+c^2=44\)
\(ab-ac-bc=8\)
\(\Rightarrow\left(a+b-c\right)^2=?\)
5.
\(x-y+z=-5\)
\(y\cdot\left(x+z\right)-7=xz\)
\(\Rightarrow x^2+y^2+z^2=?\)
6.
\(\samefrac1x+\samefrac1y+\samefrac1z=\samefrac{12}{xyz}\)
\(x+y+z=7\)
\(\Rightarrow x^2+y^2+z^2=?\)
7.
\(x,y,z\in\mathbb R^+\)
\(x^2+2xy=6\)
\(y^2+2yz=10\)
\(z^2+2xz=9\)
\(\Rightarrow x+y+z=?\)
8.
\(\samefrac1x+\samefrac1y+\samefrac1z=\samefrac8{xyz}\)
\(x+y+z=6\)
\(\Rightarrow x^2+y^2+z^2=?\)
9.
\(a+b+c=5\)
\(ab+bc+ac=3\)
\(\Rightarrow a^2+b^2+c^2=?\)
10.
\(x,y,z\in\mathbb R^-\)
\(x^2+2xy=9\)
\(y^2+2yz=11\)
\(z^2+2zx=16\)
\(\Rightarrow x+y+z=?\)
11.
\(x+y+z=4\)
\(xy+yz+xz=2\)
\(\Rightarrow x^2+y^2+z^2=?\)
12.
\(x-y+z=3\)
\(xy-xz+yz=4\)
\(\Rightarrow x^2+y^2+z^2=?\)