After isolating one square root, you will have something like $\sqrt{\text{expression}} = 2 + \sqrt{\text{another expression}}$. When you square both sides, remember that $(a+b)^2 = a^2 + 2ab + b^2$, not just $a^2 + b^2$.

Once you square and simplify, you should end up with an equation that has a single square root on one side and a regular number on the other, like $\sqrt{x+5} = \text{(number)}$. How can you solve that kind of equation?

Because you squared the equation, extraneous (false) solutions are possible. After you find $x$, substitute it back into $\sqrt{x+21} - \sqrt{x+5}$ to make sure the result is actually 2.

In Desmos, enter y = sqrt(x+21) - sqrt(x+5) on one line and y = 2 on another line. Make sure the viewing window includes x-values greater than or equal to -5 so the square roots are defined.

Look for the point where the curve $y = \sqrt{x+21} - \sqrt{x+5}$ intersects the horizontal line $y = 2$. Click on the intersection; the x-coordinate of this point is the value of $x$ that solves $h(x) = 2$.

We are given

and we want $h(x)=2$, so we solve

Because of the square roots, the expressions inside them must be nonnegative, so $x+21 \ge 0$ and $x+5 \ge 0$, which together mean $x \ge -5$ (already given). Any solution we find must also satisfy this and must work in the original equation (we will check at the end).

To eliminate square roots by squaring, first isolate one radical term. Add $\sqrt{x+5}$ to both sides:

Now we have one square root by itself on the left, which is ready to be squared.

Square both sides of

The left side is simple:

For the right side, use $(a+b)^2 = a^2 + 2ab + b^2$ with $a=2$ and $b=\sqrt{x+5}$:

So the equation becomes

Combine like terms on the right:

Subtract $x$ from both sides to eliminate $x$:

Subtract $9$ from both sides:

Now divide both sides by $4$:

Now remove the remaining square root by squaring both sides of

Squaring gives

So

Finally, check in the original equation $h(x) = \sqrt{x+21} - \sqrt{x+5}$:

• $h(4) = \sqrt{4+21} - \sqrt{4+5} = \sqrt{25} - \sqrt{9} = 5 - 3 = 2$.

The equation $h(x)=2$ is satisfied and $x=4 \ge -5$, so the solution is $x=4$.