How do you factor #a^2+4ab-3b^2 #?

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Answer 1 · 2015-06-28T17:04:25

Complete the square to find:

#a^2+4ab-3b^2 = (a+(2-sqrt(7))b)(a+(2+sqrt(7))b)#

#(a+2b)^2 = a^2+4ab+4b^2#

So

#a^2 + 4ab - 3b^2#

#= (a+2b)^2 - 4b^2 - 3b^2#

#= (a+2b)^2 - 7b^2#

#= (a+2b)^2 - (sqrt(7)b)^2#

#= ((a+2b)-sqrt(7)b)((a+2b)+sqrt(7)b)#

#= (a+(2-sqrt(7))b)(a+(2+sqrt(7))b)#

...using the difference of squares identity

#A^2-B^2 = (A-B)(A+B)#

with #A=a+2b# and #B=sqrt(7)b#