Find two consecutive integers whose product is two more than twice the sum of the integers.

Question

Written in algebra form. 

Answer 1

Let the two consecutive integers be x and x+1.  Their product (x*(x+1)) is two more than twice the sum (2*(x+x+1) + 2) of the integers:

 

     x*(x+1) = 2*(x+x+1) + 2

     x² + x = 2*(2x+1) + 2

     x² + x = 4x + 4

     x² - 3x - 4 = 0

     (x-4)(x+1) = 0

 

     x = 4 and -1

 

So there are two pairs of consecutive integers {x, x+1} that meet the criteria: {4,5} and {-1,0}.  Check:

 

     4*5 = 2*(4+5) + 2

     20 = 18 + 2

     20 = 20

     Check!

 

    -1*0 = 2*(-1+0) + 2

     0 = -2 + 2

     0 = 0

     Check!