What are two consecutive numbers whose cube differs by 271

Two consecutive numbers cubes differ by 271. Let the two consecutive numbers be x and x + 1. The difference of the two consecutive numbers is 271. ( x + 1 ) ^3 - x ^3 = 271. 1 + 3 x ^ 2 + 3 x - 271 = 0.

Step-by-step explanation:

9514 1404 393

  9, 10  or  -10, -9

Step-by-step explanation:

Let x represent the smaller of the two numbers. Then x+1 is the larger, and their difference is ...

  (x +1)³ -x³ = 271

  x³ +3x² +3x +1 -x³ = 271

  3x² +3x -270 = 0 . . . . . . simplify

  x² +x -90 = 0 . . . . . . . . divide by 3

  (x +10)(x -9) = 0 . . . . . factor

The two solutions are x = -10, x = 9.

The numbers could be {-10, -9} or {9, 10}.

Additional comment

Solutions to a factored polynomial are the values of the variable that make the factors zero. The product will only be zero if a factor is zero.

The factor (x -a) is zero when x=a.

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