Question

One integer is nine more than two times another integer. If the product of the integers is 18, how do you find the two integers?

Answer 1

Solutions integers: `color(blue)(-3,-6)`

Explanation:

Let the integers be represented by `a` and `b`.
We are told:
[1]`color(white)("XXX")a=2b+9` (One integer is nine more than two time the other integer)
and
[2]`color(white)("XXX")a xx b = 18` (The product of the integers is 18)

Based on [1], we know we can substitute `(2b+9)` for `a` in [2];
giving
[3]`color(white)("XXX")(2b+9) xx b=18`

Simplifying with the target of writing this as a standard form quadratic:
[5]`color(white)("XXX")2b^2+9b=18`

[6]`color(white)("XXX")2b^2+9b-18=0`

You could use the quadratic formula to solve for `b` or recognize the factoring:
[7]`color(white)("XXX")(2b-3)(b+6)=0`
giving solutions:
`color(white)("XXX")b=3/2` which is not permitted since we are told the values are integers.
or
`color(white)("XXX")b=-6`

If `b=-6` then based on [1]
`color(white)("XXX")a=-3`