How do you factor #y= 8m^2 - 41m - 42#?
3 Answers
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
The split of
That leads to the following possibilities to consider:
#16xx21#
#bb (48xx7)#
#112xx3#
#336xx1#
Having found the pair
#8m^2-41m-42#
#=8m^2-48m+7m-42#
#=(8m^2-48m)+(7m-42)#
#=8m(m-6)+7(m-6)#
#=(8m+7)(m-6)#
Explanation:
You could look for values
(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:
The numbers involved are still ugly but if you use a calculator or spreadsheet (evaluating only the
you should get:
Therefore one of the factors will be:
Simple division (
gives the other term:
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
Ouch!