How do you factor completely # a^2 - 10ab + 3b^2#?

1 Answer
George C.
Jan 30, 2017

#a^2-10ab+3b^2 = (a-(5+sqrt(22))b)(a-(5-sqrt(22))b)#

Explanation:

To factor this, complete the square then use the difference of squares identity, which can be written:

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

We use this with #A=(a-5b)# and #B=sqrt(22)b# as follows:

#a^2-10ab+3b^2 = a^2-10ab+25b^2-22b^2#

#color(white)(a^2-10ab+3b^2) = a^2-2a(5b)+(5b)^2-22b^2#

#color(white)(a^2-10ab+3b^2) = (a-5b)^2-(sqrt(22)b)^2#

#color(white)(a^2-10ab+3b^2) = ((a-5b)-sqrt(22)b)((a-5b)+sqrt(22)b)#

#color(white)(a^2-10ab+3b^2) = (a-(5+sqrt(22))b)(a-(5-sqrt(22))b)#

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