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

1 Answer
Jun 28, 2015

Answer:

Complete the square to find:

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

Explanation:

#(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#