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 $x = 1-sqrt(7)$

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