Sabtu, 18 Juni 2011

Math Problems

For a, b, c are constants. solve the equation system with the fastest way
x+z=a;y+z=b;xy+yz+zx=c

Solutions
Multiplying the two first equations and subtracting then the third, we get z^2=ab-c and so :

If ab<c : no real solution
If ab=c : a unique solution (x,y,z)=(a,b,0)
If ab>c : two real solutions (x,y,z)\in\left\{(a-\sqrt{ab-c},b-\sqrt{ab-c},\sqrt{ab-c}),(a+\sqrt{ab-c},b+\sqrt{ab-c},-\sqrt{ab-c})\right\}
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