MathLabs

Problem 4

Find all sets of four real numbers x1,x2,x3,x4x_1,x_2,x_3,x_4 such that the sum of any one number and the product of the other three is equal to 22.
Step 2 of 5: Derive the common quadratic
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t=x1x2x3x4,qquadxi+txi=2Longrightarrowxi2−2xi+t=0.t=x_1x_2x_3x_4,\\qquad x_i+\frac{t}{x_i}=2\\Longrightarrow x_i^2-2x_i+t=0.
Detailed analysis

The product of the three numbers other than xix_i is t/xit/x_i. The condition therefore gives xi+t/xi=2x_i+t/x_i=2, or xi2−2xi+t=0x_i^2-2x_i+t=0. Thus all four numbers are roots of one quadratic.