How do you factor #y= 8m^2 - 41m - 42#?

3 Answers
Dec 27, 2015

Answer:

Find a suitable splitting of the middle term, then factor by grouping to find:

#8m^2-41m-42=(8m+7)(m-6)#

Explanation:

Find a pair of factors of #AC = 8*42 = 16*21 = 336# which differ by #B=41#

The split of #336# into a pair of factors must put all of the powers of #2# on one side, since the difference (#41#) is odd. If both factors were even, then the difference would be even too.

That leads to the following possibilities to consider:

#16xx21#

#bb (48xx7)#

#112xx3#

#336xx1#

Having found the pair #48#, #7# use that to split the middle term and factor by grouping:

#8m^2-41m-42#

#=8m^2-48m+7m-42#

#=(8m^2-48m)+(7m-42)#

#=8m(m-6)+7(m-6)#

#=(8m+7)(m-6)#

Dec 27, 2015

Answer:

#y=(8x+7)(x-6)#

Explanation:

You could look for values #p, q, r, s# such that
#color(white)("XXX")pxxr=8#
#color(white)("XXX")qxxs=-42#
#color(white)("XXX")ps+qr=-41#
(perhaps using the AC method)

...but unless you get lucky, there are quite a few factorings possible.

As an alternative you could use the quadratic formula:
#color(white)("XXX")(-b+-sqrt(b^2-4ac))/(2a)#

The numbers involved are still ugly but if you use a calculator or spreadsheet (evaluating only the #+# of the #+-#)
you should get:
#color(white)("XXX")x=6# as a zero for this expression.

Therefore one of the factors will be:
#color(white)("XXX")(x-6)#
Simple division (#8x^2divx=8#) and (#(-42)div(-6))=+7#)
gives the other term:
#color(white)("XXX")(8x+7)#

Dec 27, 2015

Answer:

Alternatively, complete the square to find:

#8m^2-41m-42 = (m-6)(8m+7)#

Explanation:

Alternatively, you can complete the square to proceed directly to the answer as follows:

#8m^2-41m-42#

#=8(m^2-41/8 m - 21/4)#

#=8(m^2-41/8 m + (41/16)^2 - (41/16)^2 - 21/4)#

#=8((m-41/16)^2 - 1681/256 - 1344/256)#

#=8((m-41/16)^2 - 3025/256)#

#=8((m-41/16)^2 - (55/16)^2)#

#=8((m-41/16) - 55/16)((m-41/16) + 55/16)#

#=8(m-96/16)(m+14/16)#

#=8(m-6)(m+7/8)#

#=(m-6)(8m+7)#

...using the difference of squares identity:

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

with #a = m-41/16# and #b = 55/16#

Ouch!