1. \( \sqrt{x}+x=0 \)

To solve the equation \( \sqrt{x} + x = 0 \), we can rearrange it to \( \sqrt{x} = -x \). Remember, the square root of a number is always non-negative, which means \( -x \) must also be non-negative. This constraint implies that \( x \) must be less than or equal to 0. The only value that fits this criteria is \( x = 0 \). Plugging that back into the original equation confirms it's a valid solution! Now, speaking of square roots, let's highlight a fun fact: Square root calculations have a long history, dating back to ancient Babylonian mathematics! The Babylonians used a method akin to what we call "Newton's method" today to find roots, showcasing just how clever our ancestors were—solving equations long before calculators existed! On a practical note, it’s essential to maintain clarity when working with square roots. A common mistake is neglecting the fact that \( \sqrt{x} \) is defined only for \( x \geq 0 \). Make sure to always look for these restrictions when solving equations to avoid dropping extraneous solutions!