# How do you simplify sqrt(3x^2 - 12x + 12)?

Jul 13, 2015

$\sqrt{3 {x}^{2} - 12 x + 12} = \sqrt{3} \cdot \left\mid x - 2 \right\mid$

#### Explanation:

When $a , b \ge 0$ we have $\sqrt{a b} = \sqrt{a} \sqrt{b}$

$\sqrt{3 {x}^{2} - 12 x + 12}$

$= \sqrt{3 \cdot \left({x}^{2} - 4 x + 4\right)}$

$= \sqrt{3 \cdot {\left(x - 2\right)}^{2}}$

Now $3 \ge 0$ and ${\left(x - 2\right)}^{2} \ge 0$ for all $x \in \mathbb{R}$

So:

$\sqrt{3 \cdot {\left(x - 2\right)}^{2}}$

$= \sqrt{3} \cdot \sqrt{{\left(x - 2\right)}^{2}}$

$= \sqrt{3} \cdot \left\mid x - 2 \right\mid$

Note that ${\left(x - 2\right)}^{2}$ has two square roots, namely $\left(x - 2\right)$ and $- \left(x - 2\right)$.

$\sqrt{{\left(x - 2\right)}^{2}}$ denotes the positive square root, so we have to pick the positive value from $\left(x - 2\right)$ and $- \left(x - 2\right)$, which we can do by using $\left\mid x - 2 \right\mid$