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 3 of 5: Handle the single-root case
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x1=x2=x3=x4=rLongrightarrowr3+r=2Longrightarrowr=1.x_1=x_2=x_3=x_4=r\\Longrightarrow r^3+r=2\\Longrightarrow r=1.
Detailed analysis

If all four numbers coincide, the original condition is r+r3=2r+r^3=2. Since r3+r−2=(r−1)(r2+r+2)r^3+r-2=(r-1)(r^2+r+2) and the quadratic factor is always positive, the only real value is r=1r=1.