# How do you simplify sqrt(m+3) - sqrt (m-1) = 1?

Mar 26, 2017

$\textcolor{g r e e n}{m = \frac{13}{4}}$

#### Explanation:

Given $\sqrt{m + 3} - \sqrt{m - 1} = 1$

separating the roots:
$\rightarrow \textcolor{w h i t e}{\text{XX}} \sqrt{m + 3} = \sqrt{m - 1} + 1$

squaring both sides
$\rightarrow \textcolor{w h i t e}{\text{XX}} m + 3 = \left(m - 1\right) + 2 \sqrt{m - 1} + 1$

combining the $\left(- 1\right)$ and $+ 1$, and subtracting $m$ from both sides
$\rightarrow \textcolor{w h i t e}{\text{XX}} 3 = 2 \sqrt{m - 1}$

reversing the sides and dividing both sides by 2
$\rightarrow \textcolor{w h i t e}{\text{XX}} \sqrt{m - 1} = \frac{3}{2}$

squaring both sides
$\rightarrow \textcolor{w h i t e}{\text{XX}} m - 1 = \frac{9}{4}$

adding $1 \left(= \frac{4}{4}\right)$ to both sides
$\rightarrow \textcolor{w h i t e}{\text{XX}} m = \frac{13}{4}$

Mar 26, 2017

$m = 3 \frac{1}{4}$

#### Explanation:

$\sqrt{m + 3} - \sqrt{m - 1} = 1$

$\therefore \sqrt{m + 3} = \sqrt{m - 1} + 1$

square L.H.S.and R.H.S.

$\therefore {\left(\sqrt{m + 3}\right)}^{2} = {\left(\sqrt{m - 1} + 1\right)}^{2}$

$\sqrt{a} \times \sqrt{a} = a$

$\therefore m + 3 = \left(m - 1\right) + 2 \sqrt{m - 1} + 1$

$\therefore m + 3 = m \cancel{- 1} + 2 \sqrt{m - 1} \cancel{+ 1}$

$\therefore m + 3 = m + 2 \sqrt{m - 1}$

$\therefore 3 = \cancel{m} \cancel{- m} + 2 \sqrt{m - 1}$

$\therefore 3 = 2 \sqrt{m - 1}$

$\therefore 2 \sqrt{m - 1} = 3$

$\therefore \sqrt{m - 1} = \frac{3}{2}$

square L.H.S.and R.H.S.

$\therefore {\left(\sqrt{m - 1}\right)}^{2} = {\left(\frac{3}{2}\right)}^{2}$

$\therefore m - 1 = \frac{9}{4}$

$\therefore m = 1 + \frac{9}{4}$

$\therefore m = \frac{13}{4}$

$\therefore m = 3 \frac{1}{4}$

check:

$\therefore \sqrt{3.25 + 3} - \sqrt{3.25 - 1} = 1$

$\therefore \sqrt{6.25} - \sqrt{2.25} = 1$

$\therefore 2.5 - 1.5 = 1$

$\therefore 1 = 1$