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 1 of 3: Use the sum and product
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In plain words

The sum and product reduce the problem to one parameter.

x+y=a−z,xy=z2x+y=a-z,\qquad xy=z^2
Detailed analysis

The first and third equations give x+y=a−zx+y=a-z and xy=z2xy=z^2, so x,yx,y are roots of t2−(a−z)t+z2=0t^2-(a-z)t+z^2=0.