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 5 of 5: List the solutions
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x3=x4=1Longrightarrowt=1Longrightarrow(1,1,1,1);qquadx3=x4=−1Longrightarrowt=−3Longrightarrowxi=3,−1,−1,−1.x_3=x_4=1\\Longrightarrow t=1\\Longrightarrow(1,1,1,1);\\qquad x_3=x_4=-1\\Longrightarrow t=-3\\Longrightarrow\\{x_i\\}=\\{3,-1,-1,-1\\}.
Detailed analysis

The positive choice gives t=1t=1 and the quadratic collapses to (X−1)2(X-1)^2, producing (1,1,1,1)(1,1,1,1). The negative choice makes −1-1 a root of X2−2X+tX^2-2X+t, so t=−3t=-3 and the other root is 33. Thus the second family is every permutation of (3,−1,−1,−1)(3,-1,-1,-1); direct substitution verifies both families.