How do you solve #(x-1)^2=7# by factoring?

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Answer 1 · 2015-08-31T05:41:36

I defer to George C. for the correct answer.

Answer 2 · 2015-08-31T06:35:53

Rearrange as a difference of squares, then use the difference of squares identity to provide the factoring, hence the roots.

#x = 1+sqrt(7)# or #x = 1-sqrt(7)#

To solve this by factoring, first subtract #7# from both sides to get:

#(x-1)^2 - 7 = 0#

Since #7 = (sqrt(7))^2#, we can write this as:

#(x-1)^2 - (sqrt(7))^2 = 0#

Now the left hand side is a difference of squares, so we can use the differences of squares identity #a^2-b^2 = (a-b)(a+b)# as follows.

Let #a = x-1# and #b = sqrt(7)#

Then:

#(x-1)^2 - (sqrt(7))^2#

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

#= (x-1-sqrt(7))(x-1+sqrt(7))#

So our original equation becomes:

#(x-1-sqrt(7))(x-1+sqrt(7)) = 0#

which has roots:

#x = 1+sqrt(7)# and #x = 1-sqrt(7)#