How do you factor x^2 - 2c - c^2?

Mar 29, 2018

${x}^{2} - c \left(2 + c\right) = \left[x + \sqrt{c \left(2 + c\right)}\right] \left[x - \sqrt{c \left(2 + c\right)}\right]$

Explanation:

We want to factor ${x}^{2} - 2 c - {c}^{2}$.

First, let's factor $- c$ from the last two terms.

${x}^{2} - c \left(2 + c\right)$

Now picture this expression as the difference of two squares. Don't see it? Well, obviously ${x}^{2}$ is the square of $x$. But what expression, when squared would give $c \left(2 + c\right)$? That would be $\sqrt{c \left(2 + c\right)}$ of course!

We know that for the difference of two squares ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$ so in this case,

${x}^{2} - c \left(2 + c\right) = \left[x + \sqrt{c \left(2 + c\right)}\right] \left[x - \sqrt{c \left(2 + c\right)}\right]$.