Question

How do you factor the expression `x^2 - 7x - 18`?

Answer 1

`(x-9)(x+2)`

Explanation:

`Step 1:`
Look for two numbers(`a` and `b`) whose product is `-18` and whose sum is `-7`.
That is, `axxb = -18` and `a+b = -2`

To find those numbers write down the factors of `-18` which are: `{-1,1,-2,2,-3,3,-6,6,-9,9,-18,18}`

Now you will need some intuition to see that `-9xx2 = -18`
and `-9+2 = -7`

So the two numbers are `-9` and `2`.

`Step 2:`
Write down the second term of the expression(`7x`) as a sum of two terms whose coeficients are the two numbers we have found above.

`x^2-7x-18 rarr x^2+2x-9x-18`

`Step 3:`
Write down the factor out the first two terms and the two other terms.

`x^2+2x-9x-18 rarr x(x+2)-9(x+2)`

`Step 4:`
You remain with two terms(`x(x+2)` and `-9(x+2)`)
which can further be factored, like this

`x(x+2)-9(x+2) rarr color(blue)((x+2)(x-9))`

Which cannot be further factored.