Problem 4

Find all sets of four real numbers

x_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

2

Step 4 of 5: Use two distinct roots

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x_1\ne x_2\\Longrightarrow x_1x_2=t,\\qquad x_3x_4=1.

Detailed analysis

If the two roots occur, choose distinct entries $x_1,x_2$ . By Vieta their product is $t$ . Since the product of all four entries is also $t$ , the remaining pair satisfies $x_3x_4=1$ . If that pair were distinct, it would also have product $t$ , forcing $t=1$ and a double root, contrary to distinctness. Hence $x_3=x_4$ and $x_3^2=1$ .