How to solve a square root equation and verify the answer

Key concept: This is a classic radical-equation setup. The algebra is short, but the final check decides whether the answer survives.

Step by step

We solve

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

Let’s isolate one radical first:

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

Now square both sides:

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

Expand the right side:

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

Subtract $x$ from both sides:

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

Subtract 81:

$-72=-18\sqrt{x}$

Divide by $-18$:

$\sqrt{x}=4$

So

$x=16$

Check the result:

$\sqrt{16+9}+\sqrt{16}=\sqrt{25}+4=5+4=9$

It works, so the solution is

$\boxed{16}$

Pitfall alert

A common mistake is to square too fast and lose track of the radical term. Keep the square root isolated as long as possible, and always verify at the end. If you skip the check, you can keep an answer that only looks right.

Try different conditions

If the right-hand side were not 9, the same method would still work: isolate one radical, square, simplify, then check. For example, changing the constant changes the final number, but not the strategy.

Further reading

radical equation, isolating radicals, verification