How do you factor #18x^2-9x-14#?

2 Answers

#(3x+2)(6x-7)#

Explanation:

The factors of 18 are (1 and 18), (-1 and -18), (2 and 9), (-2 and -9), (3 and 6), and (-3 and -6).

So, it could either be:
#(x+a)(18x+b)#,
#(-x+a)(-18x+b)#,
#(2x+a)(9x+b)#,
#(-2x+a)(-9x+b)#,
#(3x+a)(6x+b)#,
#(-3x+a)(-6x+b)#.

The factors of -14 are (1 and -14), (-1 and 14), (-2 and 7), or (2 and -7)

Now we need to combine a set of factors of -14 and 18 that would add to give -9.

These are:
-14 - (-2 and 7)
18 - (-3 and -6)

#(-2*-6)+(-3*7) = 12-21 = -9#

So, #18^2-9x-14-=(-3x-2)(-6x+7)#.

Dividing a negative out of each bracket gives the factors as:

#(3x+2)(6x-7)#

Jun 27, 2017

#(6x-7)(3x+2)#

Explanation:

Find factors of #18 and 14# whose products differ by 9.

This involves a bit of trial and error to find them.

#" "18" "14#
#" "darr" "darr#

#" "6" "7 " "rarr 3xx7 = 21#
#" "3" "2" "rarr 6xx2 = ul12#
#color(white)(wwwwww...wwwwwwwww)9#

These are the correct factors, now look at the signs.
The difference needs to be #-9#

#" "6" "-7 " "rarr 3xx-7 = -21#
#" "3" "+2" "rarr 6xx+2 = +ul12#
#color(white)(wwwwww...wwwwwwwwwwwww)-9#

This gives the factors as:

#(6x-7)(3x+2)#

An alternative method is to multiply #18 and 14# and find factors which differ by #9#.

#18 xx14 = 252#

Find factors which are close to the square root of #252#
#sqrt252 = 15.87#

A difference of #9# means that there will be about #4# or #5# either side of #15.87#
A bit of trial and error results in #12 xx21 = 252#
This gives us a clue as to the factors we need to work towards.