MathLabs

Problem 1

Solve x+y+z=a,x2+y2+z2=b2,xy=z2x+y+z=a,\quad x^2+y^2+z^2=b^2,\quad xy=z^2 in real numbers. Find the necessary and sufficient condition on aa and bb for x,y,zx,y,z to be distinct positive numbers.
Step 2 of 3: Find all possible values of z
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In plain words

The sum-of-squares equation becomes a quadratic equation for the remaining variable z.

z=−a±2a2−b2,2a2−b2≥0z=-a\pm\sqrt{2a^2-b^2},\qquad 2a^2-b^2\ge0
Detailed analysis

Using (x+y)2=x2+y2+2xy(x+y)^2=x^2+y^2+2xy gives (a−z)2−2z2=b2(a-z)^2-2z^2=b^2, or z2+2az+b2−a2=0z^2+2az+b^2-a^2=0. Thus every real solution has z=−a±2a2−b2z=-a\pm\sqrt{2a^2-b^2}, with the radicand nonnegative.