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

Aug 31, 2015

I defer to George C. for the correct answer.

Explanation:

Aug 31, 2015

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 $x = 1 - \sqrt{7}$

Explanation:

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

${\left(x - 1\right)}^{2} - 7 = 0$

Since $7 = {\left(\sqrt{7}\right)}^{2}$, we can write this as:

${\left(x - 1\right)}^{2} - {\left(\sqrt{7}\right)}^{2} = 0$

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

Let $a = x - 1$ and $b = \sqrt{7}$

Then:

${\left(x - 1\right)}^{2} - {\left(\sqrt{7}\right)}^{2}$

$= {a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

$= \left(x - 1 - \sqrt{7}\right) \left(x - 1 + \sqrt{7}\right)$

So our original equation becomes:

$\left(x - 1 - \sqrt{7}\right) \left(x - 1 + \sqrt{7}\right) = 0$

which has roots:

$x = 1 + \sqrt{7}$ and $x = 1 - \sqrt{7}$