Question

How do you factor `9q ^ { 2} - 7q - 18`?

Answer 1

`9q^2-7q-18`

Well, `7` is a prime number, so we can't factor any of the coefficients (`9, -7, -18`)

What we need to do is find two numbers that multiply to `-162` (`9 xx -18`) and adds to `-7`:

`color(white)(0)xx` `162` let's leave the sign out of it for now
`color(white)(0)+` `7`
............
`1 xx 162`
`2 xx 81`
`3 xx 54`
`6 xx 27`
`9 xx 18`

Well, that didn't work. We should check to make sure we even can factor this.

To do that, we need to graph the equation and see the roots (`x`-intercepts)

graph{y=9x^2-7x-18}

So this does have `x`-intercepts, but the roots are pretty weird numbers, with crazy decimals. That means we can't factor it.
However, we can still find solutions, using the quadratic formula:

`-b/(2a) +- (sqrt(b^2 - 4 a c))/(2a)`

`color(red)(a) = color(red)(9)`

`color(blue)(b) = color(blue)(-7)`

`color(green)(c) = color(green)(-18)`

`(-(-7))/(2(9)) +- (sqrt((-7)^2 - 4 xx (9) xx (-18)))/(2(9))`

`7/18 +- (sqrt(49 - 4 xx (9) xx (-18)))/18`

`7/18 +- sqrt(49- -648)/18`

`7/18 +- sqrt(697)/18`

`(7 +- sqrt697)/18`

We can't simplify `sqrt(697)`, because it simplifies to `17 xx 41`, both of which are prime.

So, we have two solutions:

number `1`

`7 + sqrt(697)/18 ~~ 18.47`

number `2`

`7 - sqrt(697)/18 ~~ 15.53`

Answer 2

can't be factored.

Explanation:

`y = 9q^2 - 7q - 18`
`D = b^2 - 4ac = 49 + 648 = 697`
Since D is not a perfect square, this trinomial can't be factored.

Answer 3

`9q^2-7q-18 = 1/36(18q-7-sqrt(697))(18q-7+sqrt(697))`

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

Use this with `a=(18q-7)` and `b=sqrt(697)` as follows:

`36(9q^2-7q-18) = 324q^2-252q-648`

`color(white)(36(9q^2-7q-18)) = (18q)^2-2(18q)7+49-697`

`color(white)(36(9q^2-7q-18)) = (18q-7)^2-(sqrt(697))^2`

`color(white)(36(9q^2-7q-18)) = ((18q-7)-sqrt(697))((18q-7)+sqrt(697))`

`color(white)(36(9q^2-7q-18)) = (18q-7-sqrt(697))(18q-7+sqrt(697))`

So:

`9q^2-7q-18 = 1/36(18q-7-sqrt(697))(18q-7+sqrt(697))`

Note that `sqrt(697)` does not simplify, since `697=17*41` has no square factors.