Question

The product of two consecutive integers is 72. The equation x(x + 1) = 72 represents the situation, where x represents the smaller integer. Which equation can be factored and solved for the smaller integer?

Answer 1

The smallest integer is 8

Explanation:

`x(x+1)=72`

applying distributive law in Right hand side and subtracting 72 from both hand sides we get

`x^2+x-72=0`

Now we are required to find the factors of 72 , such that their difference is 1. The factors are 8 and 9 and 9-8=1. Hence they satisfies our requirements. Now we split the middle term like

`x^2+9x-8x-72=0`

taking x and 8 as GCF from brackets

`x(x+9)-8(x+9)=0`

`(x-8)(x+9)=0`

Hence

if
`(x-8)=0 ; x=8`

if
`(x+9)=0 ; x=-9`

Hence our answer is 8 and 9

Answer 2

For this case we have the following equation:

`x (x + 1) = 72`
We can rewrite the expression.
For this, we first make distributive property:

`x ^ 2 + x = 72`
Then, we add -72 on both sides of the equation:

`x ^ 2 + x - 72 = 72 - 72`
Rewriting we have:

`x ^ 2 + x - 72 = 0`
This is the equation that must be factored and solved for the smallest integer.
Answer:
An equation that can be factored and solved for the smaller integer is:

`x ^ 2 + x - 72 = 0`