Question

How do you factor `-2x ^ { 2} + 7x + 9`?

Answer 1

Factor the trinomial into two binomials
`( -2x + 9) * ( x + 1 )`

Explanation:

The `7x` shows that the positive factors must be greater than the negative factors.

-2 can be factored into `-1 xx 2, or -2 xx 1`

+9 can be factored into `3 xx 3 or 9 xx 1`

The combination that works is `+ 1 xx 9` and `-2 xx 1` so

`(-2x + 9)*( x +1 )`

Answer 2

`(-2x+9)(x+1)`

Explanation:

First, make sure the expression is in the standard form of a quadratic trinomial:

a`x^2` + bx + c

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

If `a!= 1`, the `ac` method is applied.

In this case, `a= -2` so we have to use the `ac` method.

What is the ac method???

The ac method is simply the value of a times the value of c

`a=- 2`

`b= 7`

`c=9`

`a*c= -18`

Now we have to find the factors that will give a product of -18 as well as a sum of 7.

`9 * -2 =-18 (ac)`
`9+(-2) = 7 (b)`

Bingo! Now that we have these numbers, we rewrite the expression substituting `9` and `-2` for b.

-2`x^2` -2x + 9x + 9 ...this is the same as the initial expression when simplified.

Now, we factor by grouping and pulling out the gcf .

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

`-2x(x+1) + 9(x+1)`

`(-2x+9)(x+1)`

Answer 3

`(9-2x)(1+x)`
`" OR " (2x-9)(-x-1)`
`" OR " -(2x-9)(x+1)`

Explanation:

It is not comfortable to find factors of an expression where the first term is negative. There are two options to avoid this problem:

OPTION 1 Re-arrange the expression.

`-2x^2 +7x + 9" "hArr" "9+7x-2x^2`

Find factors of `9 and 2` which combine to give a difference of `7`

`color(white)(........)9" "2`
`color(white)(.......)darr" "darr`
`color(white)(........)9" "2" "rarr 1 xx -2 =-2`
`color(white)(........)1" "1" "rarr 9 xx +1 =ul+9`
`color(white)(...................................................m)+7" "larr`difference is +7

`=(9-2x)(1+x)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
OPTION 2 divide `-1` out as a factor

`-2x^2 +7x + 9" "hArr" "-(2x^2-7x-9)`

`=-(2x-9)(x+1)`

This can be written as

`=-(2x-9)(x+1)`

OR `(-2x+9)(x+1) = (9-2x)(1+x)`

OR `(2x-9)(-x-1)`