Solve an equation with square roots by checking for extraneous solutions

Key takeaway: Radical equations often look simple once you square both sides, but that step can introduce answers that do not actually work. The check at the end matters.

We solve

$-\sqrt{x+9}=9-\sqrt{x}$

First, move the radical terms so the equation is easier to manage:

$\sqrt{x}=9+\sqrt{x+9}$

At this point, notice something important: the right-hand side is always at least 9, while the left-hand side is nonnegative but usually much smaller. So we should test whether any real solution is even possible.

If we square the original form, we must be careful:

$(-\sqrt{x+9})^2=(9-\sqrt{x})^2$

That gives

$x+9=81-18\sqrt{x}+x$

Simplify:

$9=81-18\sqrt{x}$ $18\sqrt{x}=72$ $\sqrt{x}=4$ $x=16$

Now check in the original equation:

Left side:

$-\sqrt{16+9}=-5$

Right side:

$9-\sqrt{16}=5$

Since $-5\ne 5$, $x=16$ is not a valid solution.

So the equation has no real solution.

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Pitfalls the pros know 👇 The biggest trap is stopping right after squaring. Squaring both sides can create fake solutions, so you always need to substitute back into the original equation. Here, the check removes the only candidate.

What if the problem changes? If the equation were instead $\sqrt{x+9}=9-\sqrt{x}$, then squaring could produce a valid answer. The sign in front of the radical changes everything, so it is worth rewriting the equation carefully before you start.

Tags: radical equation, extraneous solution, square root