# sqrt (t) = sqrt (t - 12) + 2? solve the radical equations, of possible.

Apr 16, 2017

THIS ANSWER IS INCORRECT. SEE THE CORRECT SOLUTION ABOVE.

#### Explanation:

Start by squaring both sides to get rid of one of the radicals, then simplify and combine like terms.
${\sqrt{t}}^{\textcolor{g r e e n}{2}} = {\left(\sqrt{t - 12} + 2\right)}^{\textcolor{g r e e n}{2}}$
$t = t - 12 + 4 \sqrt{t - 12} + 4$
$t = t - 8 + 4 \sqrt{t - 12}$

Then operate on both sides of the equation to isolate the other radical.
$t \textcolor{g r e e n}{- t} = \textcolor{red}{\cancel{\textcolor{b l a c k}{t}}} - 8 + 4 \sqrt{t - 12} \textcolor{red}{\cancel{\textcolor{g r e e n}{- t}}}$
$0 \textcolor{g r e e n}{+ 8} = \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{-} 8}}} + 4 \sqrt{t - 12} \textcolor{red}{\cancel{\textcolor{g r e e n}{+ 8}}}$
color(green)(color(black)8/4)=color(green)((color(red)cancelcolor(black)4color(black)sqrt(t-12))/color(red)cancelcolor(green)4
$8 = \sqrt{t - 12}$

And square both sides again to get rid of the other radical.
${8}^{\textcolor{g r e e n}{2}} = {\sqrt{t - 12}}^{\textcolor{g r e e n}{2}}$
$64 = t - 12$

Finally, add $12$ to both sides to isolate $t$.
$64 \textcolor{g r e e n}{+ 12} = t \textcolor{red}{\cancel{\textcolor{b l a c k}{- 12}}} \textcolor{red}{\cancel{\textcolor{g r e e n}{+ 12}}}$
$76 = t$
$t = 76$

When working with radicals, always check your solutions to make sure they aren't extraneous (make sure they don't cause there to be a square root of a negative number). In this case both $76$ and $76 - 12$ are positive, so $76$ is a valid solution for $t$.

Apr 17, 2017

$x \in \left\{16\right\}$

#### Explanation:

Rearrange the equation:

$\sqrt{t} - 2 = \sqrt{t - 12}$

Square both sides:

${\left(\sqrt{t} - 2\right)}^{2} = {\left(\sqrt{t - 12}\right)}^{2}$

$t - 4 \sqrt{t} + 4 = t - 12$

Simplify:

$16 = 4 \sqrt{t}$

$4 = \sqrt{t}$

Square both sides once more.

$16 = t$

Check the solution is accurate.

sqrt(16) = sqrt(16 - 12) + 2 -> 4 = 4 color(green)(√)

Hopefully this helps!