Find two consecutive numbers whose cubes differ by 271

Question

Math word problems are very hard and I'm not good at math can you help?

Answer 1

Given

Two consecutive numbers  cubes differ by 271.

Let  the two consecutive numbers be x and x + 1.

The cubes of two consecutive numbers are x ^3 and ( x + 1 ) ^3

The difference of the two consecutive numbers is 271.

( x + 1 ) ^3 - x ^3 = 271.

[ x  ^3 + 1 + 3 x  ( x + 1 ) ] - x ^3 = 271    [ Since ( a + b ) ^3 =  a ^3 + b ^3 + 3ab (a + b) ]

x  ^3 + 1 + 3 x ^ 2 + 3 x  -  x ^3 = 271

1 + 3 x ^ 2 + 3 x  = 271

1 + 3 x ^ 2 + 3 x - 271 = 0.

3 x ^ 2 + 3 x - 270= 0.

3(  x ^ 2 +  x - 90 ) = 0.

x ^ 2 +  x - 90  = 0.

x ^ 2 + 10 x - 9x  -   90  = 0.

x ( x + 10 ) - 9 ( x + 10) = 0

( x + 10 )  ( x - 9 ) = 0

Apply zero property.

( x + 10 )   = 0  or ( x - 9 ) = 0

x = - 10  or  x =  9

Therefore x = 9.

x + 1 = 9 + 1 = 10.

Solution : The two consecutive numbers are 9 and 10.