A positive integer squared plus 6 times its consecutive integer is equal to 78. Find the integers

Question

Translate to a quadratic equation

Answer 1

We can write the first integer as n to start. That means that its consecutive integer would be written as (n+1). The problem tells us "a positive integer squared" (n² ) "plus 6 times its consecutive integer" (6(n+1) ) is 78. This produces an equation that can be factored (or plugged into the quadratic formula) to solve for the values of n that make it work.

 

n²+6(n+1)=78

 

n²+6n+6=78

 

n²+6n-72=0

 

(n+12)(n-6)=0

 

n={-12, 6}.

 

The question explicitly states that the integer is positive, so we reject the -12 and say that the two consecutive integers are 6 and 7.