How do you subtract #\frac { 5m + 25} { 2m ^ { 2} + 13m + 15} - \frac { 10m - 20} { m ^ { 2} - 4}#?

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Answer 1 · 2018-03-03T21:06:50

#-(5(m+4))/(2m^2+7m+6)#

#2m^2 + 13m + 15# can be factorised by grouping:

#2 * 15 = 30#

#10 + 3 = 13#

#10 * 3 = 30#

#2m^2 + 13m + 15 = 2m^2 + 10m + 3m + 15#

#= 2m(m+5) + 3(m+5)#

#= (2m+3)(m+5)#

#2m^2 + 13m + 15= (2m+3)(m+5)#

#5m + 25 = 5(m+5)#

#10m - 20 = 10(m - 2)#

#m^2 - 4 = (m+2)(m-2)#

with factorised expressions:

#(5(m+5))/((2m+3)(m+5)) - (10(m-2))/((m+2)(m-2))#

#=(5cancel((m+5)))/((2m+3)cancel((m+5))) - (10cancel((m-2)))/((m+2)cancel((m-2)))#

#=5/((2m+3)) - 10/((m+2))#

to subtract these two fractions, find a common denominator between them:

#(2m+3)(m+2) = 2m^2+7m+6#

#5/((2m+3)) - 10/((m+2)) = (5(m+2))/(2m^2+7m+6) - (10(2m+3))/(2m^2+7m+6)#

#(5m+10-20m-30)/(2m^2+7m+6)#

#=(-15m-20)/(2m^2+7m+6)#

or #-(5(m+4))/(2m^2+7m+6)#